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Audio Engineering Society Convention Paper
音频工程学会大会论文

Presented at the Convention
在 大会上发言2016 September 29 - October 2, Los Angeles, CA, USA
2016年9月29日至10月2日，美国加利福尼亚州洛杉矶市

This paper was peer-reviewed as a complete manuscript for presentation at this convention. This paper is available in the AES
这篇论文的完整手稿已通过同行评审，将在本次大会上发表。本文可在 AES
E-Library (http://www.aes.org/e-lib) all rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.
电子图书馆 ( http://www.aes.org/e-lib) 保留所有权利。未经《音频工程学会杂志》直接许可，不得复制本文或其中任何部分。

Force Factor Modulation in Electro Dynamic Loudspeakers
电子动态扬声器中的力因子调制

Lars Risbo , Finn T. Agerkvist , Carsten Tinggaard , Morten Halvorsen , and Bruno Putzeys
Lars Risbo , Finn T. Agerkvist , Carsten Tinggaard , Morten Halvorsen , and Bruno Putzeys Purifi, Denmark/Belgium
普里菲，丹麦/比利时 Acoustic Technology, DTU-Elektro, Technical University of Denmark, DK-2800 Lyngby, Denmark
声学技术，丹麦技术大学 DTU-Elektro, DK-2800 Lyngby, 丹麦 PointSource Acoustics, Roskilde, Denmark
PointSource Acoustics，丹麦罗斯基勒

Correspondence should be addressed to Lars Risbo (lars@purifi. dk)
通讯作者：Lars Risbo (lars@purifi. dk)

Abstract 摘要

The relationship between the non-linear phenomenon of 'reluctance force' and the position dependency of the voice coil inductance was established in 1949 by Cunningham, who called it 'magnetic attraction force'.
1949 年，坎宁安确定了 "磁阻力 "这一非线性现象与音圈电感位置相关性之间的关系，并将其称为 "磁吸引力"。
This paper revisits Cunningham's analysis and expands it into a generalised form that includes the frequency dependency and applies to coils with non-inductive (lossy) blocked impedance.
本文重温了坎宁安的分析，并将其扩展为一种包含频率相关性的通用形式，适用于具有非电感（有损）阻塞阻抗的线圈。
The paper also demonstrates that Cunningham's force can be explained physically as a modulation of the force factor which again is directly linked to modulation of the flux of the coil.
论文还证明，坎宁安力可以物理解释为力因子的调制，而力因子的调制又与线圈磁通量的调制直接相关。
A verification based on both experiments and simulations is presented along discussions of the impact of force factor modulation for various motor topologies. Finally, it is shown that the popular
报告基于实验和模拟进行了验证，并讨论了力因数调制对各种电机拓扑结构的影响。最后，还显示了流行的
This paper revisits Cunningham's analysis and expands it into a generalised form that includes the frequency dependency and applies to coils with non-inductive (lossy) blocked impedance.
本文重温了坎宁安的分析，并将其扩展为一种包含频率相关性的通用形式，适用于具有非电感（有损）阻塞阻抗的线圈。
The paper also demonstrates that Cunningham's force can be explained physically as a modulation of the force factor which again is directly linked to modulation of the flux of the coil.
论文还证明，坎宁安力可以物理解释为力因子的调制，而力因子的调制又与线圈磁通量的调制直接相关。
A verification based on both experiments and simulations is presented along discussions of the impact of force factor modulation for various motor topologies. Finally, it is shown that the popular coil impedance model does not correctly predict the force unless the new analysis is applied.
报告基于实验和模拟进行了验证，并讨论了力因数调制对各种电机拓扑结构的影响。最后表明，除非采用新的分析方法，否则常用的 线圈阻抗模型无法正确预测力。

1 Introduction 1 引言

The electro-dynamic loudspeaker recently celebrated its centenary and has not materially changed construction since the direct radiating speaker by Rice and Kellogg in 1923.
最近，电动扬声器迎来了它的百年诞辰，自 1923 年赖斯和凯洛格推出直接辐射扬声器以来，它的结构并没有实质性的改变。
Production of loudspeakers today numbers in billions of units per year and thanks to digital audio storage and distribution the speaker is by orders of magnitude the most non-linear device in the audio chain.
如今，扬声器的年产量已达数十亿只，而由于数字音频的存储和传播，扬声器在数量级上已成为音频链中最非线性的设备。
Unfortunately, the in-depth understanding and modelling of speakers has progressed at a modest pace and leaves still much work to be done.
遗憾的是，对扬声器的深入了解和建模工作进展缓慢，仍有许多工作要做。
Production of loudspeakers today numbers in billions of units per year and thanks to digital audio storage and distribution the speaker is by orders of magnitude the most non-linear device in the audio chain.
如今，扬声器的年产量已达数十亿只，而由于数字音频的存储和传播，扬声器在数量级上已成为音频链中最非线性的设备。
Unfortunately, the in-depth understanding and modelling of speakers has progressed at a modest pace and leaves still much work to be done.
遗憾的是，对扬声器的深入了解和建模工作进展缓慢，仍有许多工作要做。
A major breakthrough was the work by Cunningham in 1949 [1] who analysed the inherently non-linear response of the motor due to magnetic effects with surprising depth of insight.
一个重大突破是坎宁安在 1949 年的研究[1]，他以惊人的洞察力分析了电机因磁效应而产生的固有非线性响应。
Aside from the position dependent force factor due to the non-homogeneous field in the gap and the linearising effect of overhung coils, Cunningham analysed 'Distortion due to magnetic attraction forces'.
除了间隙中的非均匀磁场和悬空线圈的线性化效应导致的与位置相关的力因子外，坎宁安还分析了 "磁吸引力导致的变形"。
He showed that 'this effect is not dependent upon the presence of a permanent field' but instead that this force was given by [1] :
他指出，"这种效应并不取决于永久场的存在"，相反，这种力是由 [1] 给出的：
Aside from the position dependent force factor due to the non-homogeneous field in the gap and the linearising effect of overhung coils, Cunningham analysed 'Distortion due to magnetic attraction forces'.
除了间隙中的非均匀磁场和悬空线圈的线性化效应导致的与位置相关的力因子外，坎宁安还分析了 "磁吸引力导致的变形"。
He showed that 'this effect is not dependent upon the presence of a permanent field' but instead that this force was given by [1] :
他指出，"这种效应并不取决于永久场的存在"，相反，这种力是由 [1] 给出的：
where is the stored magnetic energy, is the coil current, is the position dependent inductance of the voice coil and is defined as the current dependent force factor. The force is proportional to the square of the current and the spatial gradient of the coil's inductance. The result is order distortion as well as force acting in the direction of highest inductance.
其中， 是存储的磁能， 是线圈电流， 是与音圈位置有关的电感， 是与电流有关的力因子。力与电流的平方和线圈电感的空间梯度成正比。其结果是 阶失真以及 作用于最高电感方向的力。
Cunningham considered only moving coil transducers, but we will show in the Appendix that the analysis is equally applicable to all 3 known motor type, namely 'moving coil', moving magnet' and 'moving iron'. All
坎宁安只考虑了动圈传感器，但我们将在附录中说明，该分析同样适用于所有三种已知电机类型，即 "动圈"、"动磁 "和 "动铁"。所有
three motor types produce a force in response to the coil current having a desired proportional component defined by a permanent force factor as well as an undesired quadratic response c.f. (1). We will follow the conventions of moving coil transducers and call the force factor
三种电机对线圈电流都会产生一个力，这个力有一个由永久力系数定义的理想比例分量，以及一个不理想的二次响应，即 c.f. (1)。我们将遵循动圈传感器的惯例，将力因数称为
三种电机对线圈电流都会产生一个力，这个力有一个由永久力系数定义的理想比例分量，以及一个不理想的二次响应，即 c.f. (1)。我们将遵循动圈传感器的惯例，将力因数称为 。这并不影响我们分析的一般性。
The flux in the coil is central to the analysis of the force as we will show and the term flux modulation due to the coil current is thus quite apt.
线圈中的磁通量是分析力的核心，我们将展示这一点，因此线圈电流引起的磁通量调制这一术语非常贴切。
Still we prefer the term force factor modulation since the force factor is the most important characteristic of a black box model of a motor while the magnetic field making up the flux is a complex 3-dimensional internal characteristic.
但我们更倾向于使用力因数调制这一术语，因为力因数是电机黑盒模型的最重要特征，而构成磁通的磁场则是复杂的三维内部特征。
Moreover, the term 'flux modulation' is sometimes used to refer to a change in permeability due to saturation of the iron [2].
此外，"通量调制 "一词有时也用来指由于铁饱和而导致的渗透率变化[2]。
The brevity of Cunningham's analysis (basically just one short paragraph) may be the reason for today's apparent confusion where 'flux-modulation' and 'reluctance force' are often treated as separate phenomena (see e.g., [3]).
坎宁安的分析简明扼要（基本上只有一小段），这可能是今天 "磁通调制 "和 "磁阻力 "经常被视为不同现象的原因（见 [3]）。
Still we prefer the term force factor modulation since the force factor is the most important characteristic of a black box model of a motor while the magnetic field making up the flux is a complex 3-dimensional internal characteristic.
但我们更倾向于使用力因数调制这一术语，因为力因数是电机黑盒模型的最重要特征，而构成磁通的磁场则是复杂的三维内部特征。
Moreover, the term 'flux modulation' is sometimes used to refer to a change in permeability due to saturation of the iron [2].
此外，"通量调制 "一词有时也用来指由于铁饱和而导致的渗透率变化[2]。
The brevity of Cunningham's analysis (basically just one short paragraph) may be the reason for today's apparent confusion where 'flux-modulation' and 'reluctance force' are often treated as separate phenomena (see e.g., [3]).
坎宁安的分析简明扼要（基本上只有一小段），这可能是今天 "磁通调制 "和 "磁阻力 "经常被视为不同现象的原因（见 [3]）。
The drive towards speakers with long strokes, full audio range and high linearity in very small form factors makes Cunningham's analysis even more important today than in 1949 when low power amplifiers dictated the use of speakers with large diaphragms and low excursion.
与 1949 年相比，如今的扬声器更需要长冲程、全音域和高线性度，而当时的小功率放大器要求使用大振膜和低偏移的扬声器。
Force factor modulation is a much greater problem today.
如今，力因数调制是一个更大的问题。
Force factor modulation is a much greater problem today.
如今，力因数调制是一个更大的问题。

1.1 Paper Structure 1.1 纸张结构

A fundamental analysis of the physics of a generalised electro-magnetic machine that serves as a basis for this paper is given in the Appendix. Throughout the paper we assume the use of linear magnetic materials.
附录中给出了作为本文基础的通用电磁机器的基本物理分析。本文始终假定使用线性磁性材料。
Section 2 generalises Cunningham's work to cover lossy coils and to include the frequency dependent dynamics of the force. Section 3 tests the theory with both measurements and Finite Element Simulations.
第 2 节对坎宁安的工作进行了概括，以涵盖有损线圈，并包括力的频率相关动态。第 3 节通过测量和有限元模拟对理论进行检验。
Section 4 discusses the new results as applied to the popular lumped parameter models for the speaker-impedance.
第 4 节讨论了应用于流行的扬声器阻抗整块参数模型的新结果。
Section 2 generalises Cunningham's work to cover lossy coils and to include the frequency dependent dynamics of the force. Section 3 tests the theory with both measurements and Finite Element Simulations.
第 2 节对坎宁安的工作进行了概括，以涵盖有损线圈，并包括力的频率相关动态。第 3 节通过测量和有限元模拟对理论进行检验。
Section 4 discusses the new results as applied to the popular lumped parameter models for the speaker-impedance.
第 4 节讨论了应用于流行的扬声器阻抗整块参数模型的新结果。

2 Generalisation of Cunninghams's 1949 Formula
2 坎宁安 1949 年公式的一般化

Equation (1) says that the force produced by the force factor modulation is the spatial gradient of the stored magnetic energy. This equation holds generally as shown by the analysis in the Appendix. Equally fundamental is that the stored magnetic energy due the coil current and its generated flux is:
公式 (1) 指出，力因数调制产生的力 是存储磁能的空间梯度。如附录中的分析所示，该等式一般成立。同样重要的是，线圈电流 及其产生的磁通量 所产生的存储磁能：
We will now use the generality of (1) and (2) to study the force when the coil is not a pure inductor but exhibits frequency dependent losses, e.g., from eddy currents. The impedance of speaker coils has been studied intensely in literature .
现在，我们将利用 (1) 和 (2) 的一般性来研究当线圈不是纯电感，而是表现出随频率变化的损耗（如涡流损耗）时的力。扬声器线圈的阻抗在文献 中得到了深入研究。
The first step is to combine (1) and (2) to express the force as a product of the current and the current dependent force factor (also found as (25) in the Appendix):
第一步是结合 (1) 和 (2)，将力表示为电流 和取决于电流的力系数 的乘积（也可在附录中找到 (25)）：
From Faraday's law, the current dependent flux can be found from the time integral of the induced voltage in the coil at a stationary (blocked) position :
根据法拉第定律，从静止（闭锁）位置线圈中感应电压的时间积分 可以求得与电流相关的磁通量 ：
where is the voltage induced in the coil, i.e., the voltage on the coil minus the voltage across its DCresistance . We now move to the -domain (Laplace domain) where to express the flux and note that the flux and induced voltage are linear responses of the current (thanks to our assumption of linear magnetic media). In the Laplace domain the time integral is replaced by a division by :
其中 是线圈中的感应电压，即线圈上的电压减去其直流电阻上的电压 。现在我们转到 域（拉普拉斯域）， 来表示磁通量 ，并注意到磁通量和感应电压是电流的线性响应（这得益于我们对线性磁介质的假设）。在拉普拉斯域，时间积分由除以 代替：
where is the blocked impedance of the coil and is the Laplace tranform of the current.
其中 是线圈的阻塞阻抗， 是电流的拉普拉斯变换。
It is now practical to define the generalised inductance :
现在可以定义广义电感 ：
Combining (3), (5) and (6) reveals that the instantaneous force factor due to the current is the coil current filtered by a transfer function:
结合 (3)、(5) 和 (6)，可以看出电流 产生的瞬时力因数是经过传递函数滤波的线圈电流：
where we defined the force factor transfer function . The current dependent instantaneous force factor is simply the coil current filtered by this transfer function.
其中，我们定义了力因数传递函数 。与电流相关的瞬时力因数 就是用该传递函数滤波后的线圈电流 。
We now have a generalisation of Cunningham's formula where the gradient of the inductance is generalised to a filter being the -gradient of the generalised inductance . The force factor transfer function represents a dynamic linear system with possible frequency dependent phase and magnitude response. It is noted that for a purely inductive coil (Cunningham's original work) we have that . In this special case, is simply a constant.
现在，我们对坎宁安公式进行了概括，将电感梯度概括为滤波器 ，即概括电感的 - 梯度 。力因数传递函数 代表一个动态线性系统，其相位和幅值响应可能与频率有关。我们注意到，对于纯电感线圈（坎宁安的原创作品）， 。在这种特殊情况下， 只是一个常数。
The current dependent force factor is the instantaneous proportionality between the current dependent force component and the current. However, as shown in the Appendix and dictated by conservation of energy, the back EMF generated in the coil in response to motion of the coil is counter-intuitively twice the factor times the velocity c.f. (27).
电流相关力系数 是电流相关力分量 与电流之间的瞬时比例关系。然而，如附录所示，根据能量守恒原理，线圈运动时在线圈中产生的反向 EMF 是 系数的两倍乘以速度 c.f. (27)。
A further and very practical consequence of the analysis is that the dynamic force factor modulation effect can simply fully be characterised by measuring (or simulating) the position dependency of the blocked impedance . It is not necessary to measure (or simulate) the modulation of the exact magnetic field to know the impact on the force and back EMF.
该分析的另一个非常实用的结果是，只需测量（或模拟）阻塞阻抗 的位置依赖性，就可以完全确定动态力因数调制效应的特征。无需测量（或模拟）精确磁场的调制，就能知道对力和反向 EMF 的影响。
A generalisation of Cunningham was attempted in [7]. However, only the real part of the generalised inductance (corresponding to the imaginary part of the impedance) was taken into consideration.
[7] 尝试对 Cunningham 进行了概括。不过，当时只考虑了广义电感的实部（对应于阻抗的虚部）。
This means that the force factor modulation caused by the position dependency of the resistive part of the coil impedance is ignored
这意味着线圈阻抗电阻部分的位置依赖性所引起的力因数调制被忽略了
This means that the force factor modulation caused by the position dependency of the resistive part of the coil impedance is ignored
这意味着线圈阻抗电阻部分的位置依赖性所引起的力因数调制被忽略了

2.1 Symptoms of BI-modulation
2.1 BI 调节的症状

. plus a contribution due to the force factor modulation of:
......，再加上由于受力系数调制而产生的贡献：
Fig. 1: Cross section of motor structure of the used 4" driver. The motor has rotation symmetry around the -axis indicated by a vertical line.
图 1：所用 4 英寸驱动器的电机结构截面图。电机围绕 轴对称旋转，垂直线表示该轴。
The force has both a DC component (caused by real part of ) and a harmonic component. The imaginary part of represents the position gradient of the coil losses (effective series resistance) and this also causes force factor modulation in the form of a harmonic but with no accompanying DC component. Force factor modulation by a low frequency tone will amplitude modulate (AM) a high frequency voice tone (a.k.a. IMD2).
力有直流分量（由 的实部引起）和 谐波分量。 的虚部表示线圈损耗（有效串联电阻）的位置梯度，这也会导致 谐波形式的力因数调制，但不伴有直流分量。低频音调的力因数调制将对高频语音音调（又称 IMD2）进行调幅（AM）。
AM modulation manifests itself as sidebands to the voice tone (at the sum and difference frequencies) at an amplitude relative to the voice tone of:
调幅调制表现为语音音调的边带（在总频和差频），幅度相对于语音音调为
AM modulation manifests itself as sidebands to the voice tone (at the sum and difference frequencies) at an amplitude relative to the voice tone of:
调幅调制表现为语音音调的边带（在总频和差频），幅度相对于语音音调为
,where is the amplitude of the bass tone current and is the linear force factor due to the permanent field.
其中 是低音电流的振幅， 是永久磁场的线性力系数。

3 Verification Using Finite Element Simulations and Measurements
3 利用有限元模拟和测量进行验证

The theoretical results in the previous sections were verified using both numerical simulations and measurements. A 4 ", driver with a motor c.f. Fig. 1 was simulated and built. It is a variant of the transducer used in earlier work [8].
通过数值模拟和测量验证了前几节的理论结果。我们模拟并制作了一个 4 英寸 驱动器，其电机功率如图 1 所示。它是早期研究 [8] 中使用的传感器的变体。

3.1 Finite-Element Simulations
3.1 有限元模拟

Infolytica's MagNet tool was used for a series of transient simulations of a voltage step on the coil for a range of blocked coil positions to
使用 Infolytica 的 MagNet 工具对 进行了一系列瞬态模拟，模拟了线圈上的电压阶跃，模拟了一系列闭锁线圈位置 至
图 2： 绘制与 和 作为参数。实线：测量值，虚线：MagNet 模拟。
图 2： 绘制与 和 作为参数。实线：测量值，虚线：MagNet 模拟。
in increments). The voltage, current and force acting on the voice coil were sampled at . An antialias filter was included in the simulation. The result was exported to Matlab for post-processing. The instantaneous force factor was found as the force divided by the current. Subtracting the initial value representing the linear force factor yields the dynamic (current-induced) force factor . Next the blocked impedance and (i.e., the current to force factor transfer function) were identified using FFT-based deconvolution.
增量）。作用在音圈上的电压、电流和力的采样时间为 。模拟中包含一个抗混叠滤波器。模拟结果输出到 Matlab 进行后处理。瞬时作用力系数是用作用力除以电流得出的。减去代表线性力因数的初始值，得到动态（电流引起的）力因数。接下来，使用基于 FFT 的解卷积法确定阻塞阻抗和（即电流到力因子的传递函数）。

3.2 Comparing Measurements and Simulations of the Blocked Impedance
3.2 比较阻塞阻抗的测量值和模拟值

A precision positioning stage [8] was used to position and hold the coil in the motor structure without membrane or suspension. A periodic noiselike stimulus was applied and current and voltage recorded with a sound card at a sampling rate. Impedance vs. frequency curves were estimated using a synchronous FFT. The measurement was repeated for a range of positions (from to in increments). The generalised inductance c.f. to (6) was calculated for both the measurements and MagNet simulation results and plotted for comparison in Fig. (2). A convincing match is seen in general. At the lowest frequency ) the MagNet result underestimates inductance due to the limited length of the transient simulation. A mechanical resonance around affects the measured result at .
使用精密定位台[8]将线圈定位并固定在电机结构中，无需薄膜或悬挂物。使用声卡以 采样率记录电流和电压。使用同步 FFT 估算阻抗与频率的关系曲线。测量在一定范围内重复进行（从 到 ，以 为增量）。根据测量结果和 MagNet 模拟结果计算出广义电感 c.f. (6)，并绘制成图 (2) 进行比较。总体而言，两者的吻合程度令人信服。在最低频率 ) 时，由于瞬态模拟长度有限，MagNet 的结果低估了电感值。 附近的机械共振影响了 的测量结果。

3.3 数值验证 使用 MagNet 模拟
3.3 数值验证 使用 MagNet 模拟

positions. From the interpolated , the generalised inductance was found using (6). This was then used to calculate the complex from (7). Fig 3 shows a convincing match between the force factor transfer function and the one obtained from the simulated actual force, both in magnitude and phase and across all positions. This strongly supports the theoretical result of (7). Some discrepancy is observed at higher frequencies around . This is because the resolution of the simulated data set is insufficient to reconstruct the sharp minimum of the inductance caused by the copper cap. At this minimum the gradient of the inductance v.s. position changes polarity. The peaks near the rest position towards and droops at both positive and negative position. Some asymmetry is noted: force factor modulation is greater at negative positions (coil inside the motor) than positive positions. This agrees with earlier findings [8].
位置。根据内插值，利用 (6) 计算出广义电感。然后根据 (7) 计算出复数。图 3 显示，力因数传递函数与模拟实际力得到的传递函数在幅度和相位上以及在所有位置上都非常吻合。这有力地证明了 (7) 的理论结果。在频率较高的 .这是因为模拟数据集的分辨率不足以重建由铜帽引起的电感的急剧最小值。在这个最小值处，电感随位置变化的梯度极性发生了变化。在静止位置附近达到峰值，而在正负位置则有所下降。我们注意到了一些不对称现象：负位置（线圈在电机内部）的力因子调制大于正位置。这与早先的研究结果一致[8]。
The rest position peaks at at . At this frequency a drive current of peak (e.g. when the driver is driven hard in a small box) gives an amplitude of which is about of the permanent of . Such error is comparable to or even greater than the typical position dependent variation of the permanent
静止位置 的峰值为 ，频率为 。在此频率下，峰值为 的驱动电流（例如在小盒子中大力驱动驱动器时）会产生 的 振幅，该振幅约为 的永久 的 。这种误差与永久磁场的典型位置变化相当，甚至更大。

3.4 Measurement of the DC Force
3.4 直流电力的测量

Force factor modulation produces a DC force in proportion to the real value of when the coil is driven by a sinusoidal current c.f. (8), i.e., AC current causes a DC force. One would readily assume this DC force always to point inwards [2] since the inductance grows when the coil is pushed into the motor.
当线圈由正弦电流驱动时，力因数调制会产生与 实际值成正比的直流力，即交流电流会产生直流力。由于线圈被推入电机时电感量会增加，因此人们很容易认为这个直流电总是向内的[2]。
A surprise prediction of our analysis is that at high frequencies a shorting device (e.g. a copper cap or shorting ring) can modify the inductance gradient to such an extent that the sign changes. When that happens the DC force propels the cone outwards. Fig.
我们分析得出的一个惊人预测是，在高频率下，短路装置（如铜帽或短路环）会改变电感梯度，以至于符号发生变化。当发生这种情况时，直流力会将锥体向外推动。图
4 shows the real and imaginary parts of
4 显示了
A surprise prediction of our analysis is that at high frequencies a shorting device (e.g. a copper cap or shorting ring) can modify the inductance gradient to such an extent that the sign changes. When that happens the DC force propels the cone outwards. Fig.
我们分析得出的一个惊人预测是，在高频率下，短路装置（如铜帽或短路环）会改变电感梯度，以至于符号发生变化。当发生这种情况时，直流力会将锥体向外推动。图
4 shows the real and imaginary parts of at the rest position for the test driver obtained from the MagNet simulations. The sign of the real part of changes around .
4 显示了从 MagNet 模拟中获得的测试驾驶员在静止位置 时 的实部和虚部。 的实部符号在 附近发生变化。
An experiment was done to measure the DC force caused by an AC current on the test driver (DUT)
测试驱动器（DUT）上的交流电流所产生的直流力进行了测量。
Fig. 3: The force factor modulation transfer function , Solid: from the simulated , Round markers: from the simulated force.
图 3：力因子调制传递函数 ，实线：来自模拟 ，圆标记：来自模拟力。
while minimizing excursion. terms generated by asymmetries in the permanent force factor and suspension compliance [2] might otherwise confound the test.
由永久力系数 和悬架顺应性 [2] 中的不对称产生的 项可能会对测试造成干扰。
For the high frequency test the moving mass naturally renders excursion negligible so the driver could simply be tested in free air.
在高频测试中，运动质量自然会使偏移忽略不计，因此只需在自由空气中对驱动器进行测试即可。
For the tests at low frequencies, AC cone movement was countered by another 4 " driver mounted in the same cabinet and driven with a signal of the same frequency but with a carefully adjusted phase and amplitude.
在低频测试中，交流振盆运动由安装在同一箱体中的另一个 4 英寸驱动器抵消，该驱动器由频率相同但相位和振幅经过仔细调整的信号驱动。
The cabinet was intentionally made slightly leaky so that again, long term DC excursion was controlled only by the free-air complince of the driver. For all experiments, the AC excursion remained below
箱体故意做成略微漏气的样子，这样，长期直流偏移也只能由驱动器的自由空气顺从性来控制。在所有实验中，交流偏移都低于
For the tests at low frequencies, AC cone movement was countered by another 4 " driver mounted in the same cabinet and driven with a signal of the same frequency but with a carefully adjusted phase and amplitude.
在低频测试中，交流振盆运动由安装在同一箱体中的另一个 4 英寸驱动器抵消，该驱动器由频率相同但相位和振幅经过仔细调整的信号驱动。
The cabinet was intentionally made slightly leaky so that again, long term DC excursion was controlled only by the free-air complince of the driver. For all experiments, the AC excursion remained below DC blocking capacitor was added in series with the voice coil.
箱体故意做成略微泄漏的样子，这样，长期直流偏移同样只能由驱动器的自由空气顺从性来控制。在所有实验中，交流偏移都低于直流偏移，音圈上串联了阻塞电容器。
Fig. 4: Real and Imaginary components of for from MagNet simulations of the 4 " test driver.
图 4： 的实部和虚部 ，来自 4 英寸测试驱动器的 MagNet 仿真。
Fig. 5: DC-excursion caused by a series of tone bursts of varying amplitude with a order low-pass filter applied. Solid=measured, Dash model fit.
图 5：在 低通滤波器的作用下，一系列不同振幅的 音脉冲串引起的直流激增。实线=测量值，虚线 模型拟合值。
Excursion was measured with a Keyence triangulating laser head and captured alongside coil current. DC free air compliance was estimated at by applying a 10 gram weight and recording displacement, permitting conversion between DC excursion and DC force.
用 Keyence 三角测量激光头测量偏移量，并捕捉线圈电流。直流自由空气顺应性是通过施加 10 克重物并记录位移来估算的， ，从而实现直流偏移和直流力之间的转换。
Fig. 5 shows the result for a series of tone bursts at different amplitudes. The recorded excursion was low-pass filtered at . For comparison the graph is overlaid with a plot of the predicted force multiplied by the estimated compliance. To be precise, the predicted force is the square of the current filtered by the same low-pass filter and scaled by a best fit constant which ideally equals . The experiment was repeated for and resulting in:
图 5 显示了一系列不同振幅的 音调脉冲串的结果。记录的偏移在 进行了低通滤波。为便于比较，图上叠加了预测力乘以估计顺应性的曲线图。准确地说，预测力是经相同低通滤波器滤波的电流的平方，并按最佳拟合常数 （理想情况下等于 ）缩放。对 和 重复实验，结果如下：
frequency 频率 best fit  最合适 simulated  仿真
The measured DC excursion is about to larger than expected. Possible contributing error factors is that is strongly dependent on the exact rest position and that the compliance varies over time.
测得的直流偏移约比预期的 至 大。可能造成误差的因素是 与确切的静止位置密切相关，而且顺应性随时间而变化。
Next a series of bursts at varying amplitudes were applied. As expected this resulted in a negative DC excursion as shown in Fig. 6. Again the best fit model is overlaid with . This is quite close to the simulated value , i.e., about the same relative error as for the high frequency bursts. However, the excursion is progressively larger than the quadratic model for large currents.
接下来是一系列不同振幅的 脉冲串。不出所料，如图 6 所示，这导致了负直流偏移。最佳拟合模型再次与 相叠加。这与模拟值 非常接近，即与高频脉冲串的相对误差大致相同。然而，在大电流情况下，偏移量会逐渐大于二次模型。
Most likely we are seeing our starting assumption, that the magnetic materials are substantially linear, become progressively more inaccurate as the drive current goes up. The slow settling time is dictated by the air leak. Finally, a
最有可能的情况是，随着驱动电流的增加，我们开始假设的磁性材料基本上是线性的，但这一假设变得越来越不准确。缓慢的沉淀时间是由漏气决定的。最后
burst was applied (with active AC excursion cancellation). This frequency was chosen since the real value of here is very close to zero. Indeed, the plot shows that the excursion settles essentially back to zero after a positive transient.
猝发（主动交流偏移消除）。选择这个频率是因为这里的实际值非常接近于零。事实上，从图中可以看出，在正瞬态之后，偏移基本上会恢复到零。

4 Lumped Parameter Models
4 整数参数模型

In literature several simplified models have been suggested to approximate the way the blocked electrical impedance of a real loudspeaker depends on frequency. Some of them, , provide an equation that directly describes the impedance of the lossy inducance
文献中提出了几种简化模型，以近似实际扬声器的阻塞电阻抗随频率变化的方式。其中一些模型，如 ，提供了一个直接描述有损电感阻抗的方程
Fig. 6: DC-excursion caused by a series of tone bursts of varying amplitude and a single burst. All plots filtered by a order lowpass filter. Solid=measured, Dash=model fit.
图 6：由一系列不同振幅的 音频脉冲串和一个 脉冲串引起的直流激增。所有图均由 阶 低通滤波器滤波。实线=测量值，虚线=模型拟合值。
and the generalized calculation method presented in this paper can be applied in a straightforward manner on these models, while other authors [11], [12] provide a lumped parameter network.
而其他作者[11]、[12]则提供了一个集合参数网络。
For the model, illustrated in Fig. 7, the current in , can be shown to be a low pass filtered version of the current in :
对于 模型（如图 7 所示）， 中的电流可以证明是 中电流的低通滤波版本：
which in the time domain becomes
在时域中变为
with and represents convolution and is the impulse response of the filter :
和 表示卷积， 是滤波器 的脉冲响应：
It is tempting to apply Cunninghams equation directly [11] on each of the two ideal inductances and their respective currents, and , which gives
我们不妨将坎宁安方程 [11] 直接应用于两个理想电感及其各自的电流 和 ，从而得出
Fig. 7: Model of blocked voice coil impedance using the model. Dashed box indicate inductive part of impedance.
图 7：使用 模型的阻塞音圈阻抗模型。虚线框表示阻抗的电感部分。
如果假设 和 力随位移变化而变化。
如果假设 和 力随位移而变化。
i.e., the current is filtered through before being squared.
也就是说，电流经过 滤波后再进行平方运算。
The generalized method of calculation from section 2 is based on the impedance of the network
第 2 节中的通用计算方法基于网络阻抗
from which the generalised inductance can be calculated as
由此可计算出广义电感为
and equations (3) and (7) gives
根据公式 (3) 和 (7) 可以得出
i.e., the current is filtered once before being multiplied with itself, and force factor transfer function is
即电流在与自身相乘之前先滤波一次，力因数传递函数为
This disagreement on the influence of the filter is in fact due to an incorrect interpretation of the model. One should see it as a black box, where the inside details cannot be trusted to give physical meaning. While the model is valid for predicting the blocked impedance, and in particular its current, , is a model abstraction, which does not exist in the physical system. The physical systems only contains one (lossy) inductance and current, , and the flux generated by and are not independent. Consequently one cannot use Cunninghams equation on directly. However if one goes back to the fundamental equation for the energy stored (3):
对滤波器 影响的这种分歧，实际上是由于对 模型的不正确解释造成的。我们应将其视为一个黑盒子，不能相信其内部细节能给出物理意义。虽然该模型对于预测阻塞阻抗是有效的，但 ，尤其是其电流 ，只是一个模型抽象，在物理系统中并不存在。物理系统只包含一个（有损耗的）电感和电流，即 ，而 和 产生的磁通量并不是独立的。因此，我们不能直接在 上使用坎宁安方程。不过，如果我们回到存储能量的基本方程 (3)，就会发现：(3)
Fig. 8: Currents in the model and force contributions for calculated directly on and via the network impedance .
图 8： 模型中的电流以及直接在 和通过网络阻抗 计算的 的力贡献。
the correct result is obtained in agreement with equation (18), indicating that the total flux ( ) is accurately explained by the model, but the additional energy stored in is not correct, because the flux generated in interacts with the flux from .
得出的正确结果与方程 (18) 一致，表明 模型可以准确解释总通量 ( ) ，但 中存储的额外能量并不正确，因为 中产生的通量与 中的通量相互作用。
The difference between the two result is illustrated in Fig. 8. Here the force originating from only is shown in the case where the current is stepped from 0.5 to 1 amperes. The generalized calculation shows a clear step in the force, whereas the calculation based on equation (14) does not and deviates significantly in the transient response.
图 8 显示了两种结果之间的差异。这里显示的是电流从 0.5 安培阶跃到 1 安培时，仅来自 的力。广义计算显示了力的明显阶跃，而基于方程 (14) 的计算则没有，并且在瞬态响应中出现了明显偏差。
The method presented here can also be applied to more advanced lumped parameter models containing more elements such as suggested in [12], so long as the model has a valid connection between the voltage drop over the inductive (excluding ) part and the total flux generated.
这里介绍的方法也可用于包含更多元素的更先进的叠加参数模型，如文献 [12] 所建议的模型，只要该模型在电感（不包括 ）部分的压降与所产生的总磁通之间存在有效的联系。

5 Conclusion 5 结论

The results of Cunningham can be generalised to comprise all electromagnetic motor/actuator types and include the general frequency dependency of the force factor transfer function for coils with losses.
坎宁安的研究结果可以推广到所有电磁电机/执行器类型，并包括带损耗线圈的力因传递函数的一般频率依赖性。
The presented framework highlights the close relationship between the position modulation of the coil's blocked impedance and the force factor modulation. The measurements and numerical simulations support the theory
所提出的框架强调了线圈阻塞阻抗的位置调制与力因子调制之间的密切关系。测量和数值模拟支持了这一理论
The presented framework highlights the close relationship between the position modulation of the coil's blocked impedance and the force factor modulation. The measurements and numerical simulations support the theory
所提出的框架强调了线圈阻塞阻抗的位置调制与力因子调制之间的密切关系。测量和数值模拟支持了这一理论
and indicate that the force factor modulation is significant and should not be ignored since it can easily be as large as , i.e., comparable to other large signal errors [2]. The underlying mechanism is that the force factor changes if a change in stored magnetic energy happens when the motor moves position.
并表明力因数调制非常重要，不应被忽视，因为它很容易达到 ，即与其他大信号误差相当[2]。其基本机理是，当电机移动位置时，如果存储的磁能发生变化，力因数也会发生变化。
This storage of energy causes an asymmetry between the force factor and the ratio between velocity and the back-EMF produced by the motor. Finally, it was concluded that the popular
这种能量储存导致力系数与速度和电机产生的反向电磁场之间的比率不对称。最后得出的结论是
impedance model does not predict the correct dynamic force when the original Cunningham equation is used on the inductor alone. Continued research must be done in expanding the analysis framework to include non-linear magnetic response.
当原始的坎宁安方程仅用于电感器时，阻抗模型无法预测正确的动态力。必须继续研究如何扩展分析框架，以纳入非线性磁响应。

Acknowledgements 致谢

The authors would like to thank Samel Arslanagic from DTU Elektro for fruitful discussions on the electromagnetic analysis and Christian Berg Jacobsen of PointSource Acoustics for preparing the speaker test assembly.
作者感谢 DTU Elektro 公司的 Samel Arslanagic 就电磁分析进行了富有成效的讨论，并感谢 PointSource Acoustics 公司的 Christian Berg Jacobsen 准备了扬声器测试组件。

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[15] Aldoshina, I. A., Voishvillo, A., and Mazin, V., "Modeling of Flux Modulation Distortion in Moving-Coil Loudspeakers by the Finite-Element Method," in Audio Engineering Society Convention 98, 1995 .
[15] Aldoshina, I. A., Voishvillo, A., and Mazin, V., "Modeling of Flux Modulation Distortion in Moving-Coil Loudspeakers by the Finite-Element Method," in Audio Engineering Society Convention 98, 1995.

A Force and Energy Balance of an Electromagnetic Machine
电磁机器的力与能量平衡

Consider an electromagnetic machine of Fig. 9 having a single coil with flux , no DC resistance and driven by a current source resulting in the coil voltage
考虑图 9 中的电磁机，它有一个单线圈，磁通量为 ，无直流电阻，由电流源 驱动，线圈电压为
Fig. 9: A generalised single coil electromagnetic machine considered for a virtual displacement at constant current . The machine acts with a force resulting in a work and consumes electrical energy whilst storing magnetic energy .
图 9：在恒定电流下考虑虚拟位移 的通用单线圈电磁机 。该机器的作用力为 ，产生功 ，消耗电能 ，同时储存磁能 。
. One or more optional permanent magnets (or energised field coils with constant current) provide a permanent magnetic field in combination with ferromagnetic materials to shape the field.
.一个或多个可选的永久磁铁（或恒定电流的通电磁场线圈）提供永久磁场，与铁磁性材料结合形成磁场。
For the purpose of this analysis we assume the magnetic materials to be linear so that the response from current to the magnetic
在本分析中，我们假设磁性材料是线性的，因此电流对磁场的响应是线性的。
-field is linear. The machine has a moving part that can move along an -axis whilst producing a force acting on its exterior. The moving part of the machine can be the coil itself, a piece of iron or a magnet. Such transducers are called moving coil, moving iron and moving magnet transducers respectively.
-场是线性的。机器有一个运动部件，可以沿某一轴线运动，同时在其外部产生作用力。机器的移动部件可以是线圈本身、一块铁或一块磁铁。这种传感器分别称为动圈、动铁和动磁传感器。
Without loss of generality a similar analysis could be made for a rotating motor defined by its shaft angle
在不失一般性的前提下，可以对旋转电机进行类似的分析，该电机由其轴角定义
Without loss of generality a similar analysis could be made for a rotating motor defined by its shaft angle and torque [14]
在不失一般性的前提下，可以对旋转电机进行类似的分析，该电机由轴角 和转矩 [14] 所定义。
A common method for finding forces in electromechanical systems is to study the energy balance during a tiny instantaneous 'virtual' displacement [14]. The system performs mechanical work equalling . The required energy may come from electrical work supplied by the source or energy already stored in the magnetic circuit. Since the displacement is instantaneous the current is constant and no energy is lost as heat in the meantime. Energy conservation requires that [14] . Deriving with respect to yields:
寻找机电系统中力的常用方法是研究微小瞬时 "虚拟 "位移过程中的能量平衡 [14]。系统所做的机械功等于 。所需的能量可能来自电源提供的电功 ，也可能来自已经储存在磁路中的能量 。由于位移是瞬时的，因此电流是恒定的，在此期间没有能量以热量形式损失。能量守恒要求 [14] 。根据 得出：
The coil flux is the sum of a current-independent flux component generated by the permanent magnet and a flux component which responds linearly to the coil current , i.e., we have . Both components are generally dependent on the position . The corresponding components of force ( and ) and induced voltage ( and ) will be treated separately while noting that the principle of linear superposition applies. Because of this, the analysis holds equally for the three major types of electromagnetic transducers (moving coil, moving iron and moving magnet).
线圈磁通量 是永磁体产生的与电流无关的磁通量分量 和磁通量分量 的总和，磁通量分量与线圈电流呈线性关系 ，即 。这两个分量通常都取决于位置 。相应的力分量 ( 和 ) 和感应电压分量 ( 和 ) 将分别处理，同时注意线性叠加原理。因此，本分析同样适用于三种主要类型的电磁传感器（动圈、动铁和动磁）。
Despite being vastly different in their physical construction, these transducer types only differ in behaviour in terms of the relative contributions of the current independent and current dependent components.
尽管在物理结构上存在巨大差异，但这些传感器类型在行为上的不同之处仅在于与电流无关的分量和与电流有关的分量的相对贡献率不同。
Despite being vastly different in their physical construction, these transducer types only differ in behaviour in terms of the relative contributions of the current independent and current dependent components.
尽管在物理结构上存在巨大差异，但这些传感器类型在行为上的不同之处仅在于与电流无关的分量和与电流有关的分量的相对贡献率不同。

A. 1 The Force and Voltage From the Permanent Field
A.1 永久磁场产生的力和电压

First, consider the components of force and voltage that arise strictly from the permanent field. Any change in flux (and hence any induced voltage ) can only occur as the result of movement. From Faraday's law of induction it follows that the electrical work is the integral of the product of the current and the induced voltage in the coil: . Combined with (21) we can express the permanent component of the force factor as the position gradient of the coil flux from the permanent field [15].
首先，考虑严格由永久磁场产生的力和电压分量。磁通量 的任何变化（以及任何感应电压 ）都只能是运动的结果。根据法拉第感应定律，电功 是线圈中电流与感应电压乘积的积分： 。结合 (21)，我们可以将力因数 的永久分量表示为来自永久磁场的线圈磁通量的位置梯度 [15]。
is the "classical" force factor which, multiplied with current, produces the force that in an ideal voice coil transducer would be the only one acting on the membrane. For all three motor types, is proportional to the strength of the permanent magnet. Moving iron motors can achieve a very high force factor but only over a short usable -range by having a short air gap. This principle was used in the first telephone receivers and is still used in hearing aids today.
是 "经典 "力系数，与电流相乘，产生的力是理想音圈换能器中唯一作用在膜片上的力。对于所有三种电机类型而言，都与永久磁铁的强度成正比。动铁电机可以达到很高的力因数，但只能通过短气隙在很短的可用范围内实现。这种原理曾用于第一台电话接收机，如今仍用于助听器。
expresses a force that is strictly position dependent (i.e., elastic) and is due to the position gradient of the stored energy in (21). Put simply it is the attraction force between magnet and iron if one of them is the moving part.
表示的是一种严格依赖于位置（即弹性）的力，是由于 (21) 中存储能量的位置梯度造成的。简单地说，它就是磁铁和铁之间的吸引力，如果其中一个是运动部分的话。
In moving coil transducers this term is therefore zero.
因此，在动圈换能器中，这个项为零。
In moving coil transducers this term is therefore zero.
因此，在动圈换能器中，这个项为零。
Faraday's law of induction gives us the induced voltage on the coil due to movement, i.e., the so-called back Electro Motive Force (EMF):
根据法拉第感应定律，我们可以得出线圈因运动而产生的感应电压，即所谓的反向电动势（EMF）：
Note the positive polarity is due to the definition of the coil voltage being seen from the electrical source in Fig. (9).
请注意，正极性是由于图 (9) 中从电子源看到的线圈电压的定义。

A. 2 The Force and Voltage from the Current Dependent Field
A.2 电流场产生的力和电压

Magnetic energy is stored as a result of its current and the resulting flux that the coil creates. Conceptually the stored energy can be found by ramping the current to zero and integrating the electrical work produced, i.e., we take the stored magnetic energy out as electrical work. This gives the very fundamental and general result [14]:
磁能 的储存是其电流 和线圈产生的磁通量 的结果。从概念上讲，可以通过将电流斜坡降为零并对所产生的电功进行积分来找到存储的能量，也就是说，我们将存储的磁能作为电功取出。这就得出了非常基本和一般的结果 [14]：
The displacement under this constant current condition may cause a change in the flux which changes the stored magnetic energy by . However, the change in flux also results in electrical work delivered by the source equal to . This means that , i.e., only half of the electrical work is stored as magnetic energy and the other half must equal the mechanical work to satisfy the energy balance expressed in (21):
在这种恒定电流条件下，位移 可能会导致磁通量发生变化 ，从而使存储的磁能发生 的变化。然而，磁通量的变化也会导致磁源输出相当于 的电功。这意味着， ，即只有一半的电功被储存为磁能，另一半必须等于机械功才能满足 (21) 所表示的能量平衡：
responding linearly to the current. The resulting force (by multiplication with the current) becomes a quadratic function of the current.
与电流成线性关系。由此产生的力（与电流相乘）成为电流的二次函数。
The induced voltage on the coil (Faraday's law):
线圈上的感应电压（法拉第定律）：
The first term of (26) is due to the so-called blocked (non-motional) impedance of the coil and the second term is caused by the motion, a.k.a. the back-EMF which by the help of (25) can be re-written to:
(26) 的第一项是由于线圈的所谓阻塞（非情态）阻抗造成的，第二项是由于运动（又称反向电磁场）造成的，在 (25) 的帮助下，可将其改写为：
Note that the forces of the permanent and current dependent cases are quite similar (22) and (25) but differ by a factor 2 . Similarly, a factor 2 difference is found for the back EMF: (23) and (27).
请注意，永久和电流相关情况下的力非常相似（22）和（25），但相差 2 倍。同样，反向 EMF 也有 2 倍的差异：(23) 和 (27)。
This factor 2 reflects the difference in storage of energy in the machine: the current dependent components of flux and force store energy during motion. This energy is drawn from the electrical source alongside that which produces the actual force.
系数 2 反映了机器中能量存储的差异：在运动过程中，与电流相关的磁通量和力分量存储了能量。这些能量与产生实际力的能量一起从电能源中提取。
No such storage of electrical energy happens for the components associated with the permanent field. This factor 2 seems also to have caused confusion in the earlier literature [15] .
而与永久磁场相关的部件则不会储存电能。早期文献 [15] 中的这一因素 2 似乎也引起了混淆。
This factor 2 reflects the difference in storage of energy in the machine: the current dependent components of flux and force store energy during motion. This energy is drawn from the electrical source alongside that which produces the actual force.
系数 2 反映了机器中能量存储的差异：在运动过程中，与电流相关的磁通量和力分量存储了能量。这些能量与产生实际力的能量一起从电能源中提取。
No such storage of electrical energy happens for the components associated with the permanent field. This factor 2 seems also to have caused confusion in the earlier literature [15] .
而与永久磁场相关的部件则不会储存电能。早期文献 [15] 中的这一因素 2 似乎也引起了混淆。
  1. AES Convention, Los Angeles, CA, USA, 2016 September 29 - October 2
    AES 大会，美国加利福尼亚州洛杉矶，2016 年 9 月 29 日至 10 月 2 日
    Page 3 of 10
    第 3 页，共 10 页
  2. AES Convention, Los Angeles, CA, USA, 2016 September 29 - October 2
    AES 大会，美国加利福尼亚州洛杉矶，2016 年 9 月 29 日至 10 月 2 日
    Page 9 of 10
    第 9 页，共 10 页
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