Principles and Design Analysis of Electroacoustic Conversion in Moving-Coil Loudspeakers

Published by IWISTAO

Speakers are indispensable components in audio systems, and their performance directly determines sound reproduction quality and listening experience. Among common speaker types, moving-coil loudspeakers are the most common. This paper explores in depth the electroacoustic conversion principle and provides detailed formula derivations and design analysis.

I. Basic Structure and Working Principle of Moving-Coil Loudspeakers

A typical moving-coil loudspeaker consists of the following key components:

• Voice coil: after energization, it generates a magnetic field and interacts with the magnetic circuit;
• Magnetic circuit system: provides a constant magnetic field;
• Diaphragm: driven by the voice coil to vibrate, pushing air to produce sound;
• Suspension system: including spider and surround, ensuring vertical motion of the voice coil while limiting lateral displacement.

The working principle of a moving-coil loudspeaker is as follows: when alternating current (audio signal) is input to the voice coil, electromagnetic induction causes the voice coil and magnetic circuit to generate an interaction force that drives the diaphragm to move back and forth; diaphragm vibration pushes air to radiate sound waves, realizing conversion from electrical energy to acoustic energy.

II. Electromagnetic Transduction Process and Formula Derivation

The electroacoustic conversion of moving-coil loudspeakers is essentially an electromagnetic energy conversion process. According to the Lorentz force law (Lorentz Force Law), the force acting on the voice coil can be expressed as:

F = B · l · i

where:

• F: electromagnetic force acting on the voice coil (unit: N)
• B: magnetic flux density in the gap magnetic field (unit: T)
• l: effective conductor length of the voice-coil winding (unit: m)
• i: current through the voice coil (unit: A)

The electromagnetic force on the voice coil drives diaphragm vibration, and the diaphragm motion equation can be described by the classical mass-spring-damper system:

m d²xdt² + R_m dxdt + Kx = F

where:

• m: equivalent mass of the loudspeaker vibration system (unit: kg)
• R_m: mechanical damping coefficient (unit: N·s/m)
• K: stiffness coefficient of the suspension system (unit: N/m)
• x: displacement of the voice-coil/diaphragm system (unit: m)

Substituting the electromagnetic force expression into the above equation yields:

m d²xdt² + R_m dxdt + Kx = Bli

This is the fundamental differential equation of loudspeaker electromechanical coupling.

III. Electrical Equivalent Impedance Model of the Loudspeaker

The loudspeaker voice coil also has electrical characteristics, which can be represented by an electrical equivalent impedance model:

The voltage-current relationship of the voice coil can be expressed as:

u(t) = R_ei(t) + L_e di(t)dt + e(t)

where:

• u(t): loudspeaker input voltage (unit: V)
• R_e: voice-coil resistance (unit: ohm)
• L_e: voice-coil inductance (unit: H)
• e(t): back electromotive force (Back EMF)

Because the vibrating voice coil cuts magnetic field lines and generates back EMF, according to Faraday's law of electromagnetic induction:

e(t) = Bl dxdt

In frequency-domain analysis, complex numbers are used:

X(ω), I(ω), U(ω)

satisfy:

U(ω) = (R_e + jωL_e)I(ω) + jωBlX(ω)

and the mechanical equation in the frequency domain is:

(jω)²mX(ω) + jωR_mX(ω) + KX(ω) = BlI(ω)

Combining the above two equations and eliminating displacement, the loudspeaker electrical input impedance expression can be obtained:

Z(ω) = R_e + jωL_e + (Bl)²R_m + j(ωm - Kω)

IV. Loudspeaker Sensitivity and Efficiency Analysis

An important loudspeaker metric, sensitivity, is defined as the sound pressure level at a specific distance under a specified input voltage, usually expressed in dB SPL:

Loudspeaker efficiency is defined as output acoustic power to input electrical power ratio:

η = P_acousticP_electric × 100%

Loudspeaker output acoustic power can be determined through the concept of diaphragm radiation acoustic impedance:

P_acoustic = 12R_rad(ω)v²

where:

• R_rad(ω): real part of the loudspeaker radiation acoustic impedance, representing resistance to acoustic radiation (unit: N·s/m)
• v: diaphragm velocity amplitude (unit: m/s)

Based on the relationship between diaphragm velocity and displacement, and the above relationship among displacement, current, and input voltage, one can further calculate loudspeaker sensitivity and efficiency in detail.

V. Design Optimization Considerations

In practical design, the following factors need to be considered comprehensively to optimize performance:

• Magnetic circuit design: increasing magnetic flux density B can increase electromagnetic conversion efficiency;
• Voice-coil design: reasonably select conductor length and wire diameter to optimize impedance matching;
• Diaphragm design: reducing mass m improves sensitivity while balancing stiffness and damping characteristics;
• Suspension system design: appropriate elastic coefficient K and damping R_m, to achieve reasonable frequency-response characteristics and stability.

VI. Summary

The electroacoustic conversion process of moving-coil loudspeakers is essentially a mechanically vibrating system driven by electromagnetic force; through systematic formula derivation and analysis, one can clearly understand the loudspeaker working principle and the influence of key design parameters. Loudspeaker design and optimization are a multivariable trade-off process requiring comprehensive consideration of electrical, magnetic-circuit, mechanical, and acoustic factors to achieve ideal sound reproduction state.

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