Modeling, FEM Analysis and Dynamic Simulation of a Moving Coil Loudspeaker
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1 Modeling, FEM Analysis and Dynamic Simulation of a Moving Coil Loudspeaker 22nd International Symposium on Power Electronics, Electrical Drives, Automation and Motion Ischia (Italy) June 2014 Ezio Santini, Sabrina Teodori DIAEE, Department of Astronautic, Electrical and Energetic Engineering SAPIENZA University of Rome, Via Eudossiana 18, Rome, Italy ezio.santini@uniroma1.it, sabrina.teodori@uniroma1.it
2 The goal: TO PROVIDE A SIMPLE AND EASY-TO-USE ALGORITHM FOR THE DETERMINATION OF THE MECHANICAL FORCE ACTING ON A LOUDSPEAKER MOVING COIL USE OF THE DETERMINED QUANTITY: INPUT TO AN ACOUSTIC ANALYSIS SOFTWARE METHODOLOGY: FEM ANALYSIS FOR PARAMETER EVALUATION MECHANICAL EQUATIONS FORCE FACTOR VS. DISPLACEMENT IN COOPERATION WITH SICA ALTOPARLANTI S.R.L. (ITALY) 2
3 A loudspeaker is: a linear motor with a small displacement range an electroacoustic transducer that produces sound in response to an electrical audio signal input (voltage / current) 3
4 How it is made? Mainly, it consists of: an annular permanent magnet a coil which is free to move into an airgap an iron structure as the pathway of magnetic circuit a plastic or paper cone 4
5 How does it work? 5
6 Outline: Calculation of the force factor (Bl) by studying the distribution of the flux density along the coil depth (FEM) Calculation of value of the coil self-inductance L (FEM) Simulation of the system by means of the Matlab tool Simulink Parametric analysis (what-if) 6
7 CASE STUDY Other dimensions WOOFER Power 120 W Range frequency 150/6000 Hz Coil material Aluminum Coil diameter 38 mm Coil turns 63 Permanent magnet external diameter 124 mm Permanent magnet internal diameter 44 mm 7
8 ANALITICAL AND ELECTRICAL MODELS DYNAMIC EQUATION OF THE SPEAKER MOBILE MASS axial displacement of the mobile coil M ms x R x speaker mobile mass ms C 1 ms x F( x, t) damping coefficient inverse of the spring force constant Lorentz force 8
9 ANALITICAL AND ELECTRICAL MODELS EQUIVALENT CIRCUIT OF THE COIL Voltage generator: represents the input signal Series resistance: represents the losses in the electrical conductors Variable inductance: the magnetic energy stored into the winding Voltage generator: represents the back emf. 9
10 ANALITICAL AND ELECTRICAL MODELS ELECTRICAL CIRCUIT EQUATION electromotive force d v Ri L( x) i Bl( x) i dt dx dt resistive voltage drop inductive voltage drops 10
11 ANALITICAL AND ELECTRICAL MODELS CAD GEOMETRICAL MODEL Notice that: geometric and magnetic symmetry axis is present Advantage in terms of: computing times improvement of the solution accuracy Coil 2D geometry suitable for FEM analysis - transverse section
12 ANALITICAL AND ELECTRICAL MODELS MATHEMATICAL MODEL OF PERMANENT MAGNETS B Br ( H H H c ) c Notice that: large air gaps allows to consider the iron as operating in the linear part of magnetic characteristic. Demagnetization B - H curve for a PM material for the easy axis (second quadrant). 12
13 FEM ANALYSIS Custom-made FEM analysis software has been used (2D FEM Software Amadeus ) The reference equation is the standard Poisson formulation of axisymmetric static magnetic fields Vector potential 1 ( ra) 1 A r r r z z J Permeability Current density 13
14 FEM ANALYSIS BOUNDARY CONDITIONS No magnetic barrier. But,notice that: Currents inputs exhibit intrinsically zero-divergence then: B is practically zero at a given distance from the sources then: Semi-circular boundary is a Dirichlet-type 14
15 FEM ANALYSIS INPUTS FOR FEM SOLUTION Total current flowing into the conductor Magnetic properties of the materials: Air Aluminum Iron Permanent magnets Magnetic permeability Differential magnetic permeability Residual flux density Br Geometry of transducer in terms of nodes and edges. 15
16 FEM ANALYSIS INPUT QUANTITIES FOR INDUCTIVE PARAMETERS EVALUATION : QUANTITIES VALUE Total current 1 A Magnetic potential boundary condition A = 0 Ceramic magnet (second quadrant) µ r =1,671 B r =0,42 Iron µ r =
17 FEM ANALYSIS RESULTS EQUIPOTENTIAL LINES IN: No-load case Load case Notice that: influence of the currents flowing into the coils on the magnetic field is truly minimal in fact, the aim is: B in the coil should not vary in the speaker operation. 17
18 PARAMETER IDENTIFICATION BY FEM Several configurations of the moving coil have to be analyzed, representing its displacement during the electromechanical energy conversion FLUX LINES IN THE COIL: in central position 2 mm displaced in vertical direction Notice that: when the coil goes out of the air gap, the flux lines are not anymore perfectly orthogonal to the coil displacement direction then the force produced is not parallel to the coil axis then there is a sound distortion 18
19 PARAMETER IDENTIFICATION BY FEM The coil has been moved by 1 mm steps, in a range that goes from 4 mm over the central position to 4 mm lower. B IN THE COIL B vs. coil depth for the central position of the coil itself. average B on the coil in different positions coil position [mm] average flux density [T] Notice that: B in the coil decreases when the coil moves out of the air gap then Lorentz force on the coil decreases 19
20 PARAMETER IDENTIFICATION BY FEM FORCE FACTOR (Bl) force factor vs coil position force factor on the coil in different coil positions ì coil position [mm] force factor [T m] ì
21 FEM ANALYSIS RESULTS 21
22 PARAMETER IDENTIFICATION BY FEM SELF-INDUCTANCE L: Inductance trend vs. coil displacement Notice that: the presence of the airgap has a smoothing effect on the inductance behavior This variation is a non-linearity for the simulation model A functional relationship between x and Bl(x) must be arrived at 22
23 SIMULATOR INTEGRATION OF DYNAMIC EQUATION M ms x R ms x 1 C ms x F(x, t ) Simulink has been used in order to build a magnetic motor simulator Dynamic equation has been integrated with Matlab The force factor function Bl(x) has been obtained by means OLS interpolation of the FEM data. In the case under investigation, such relationship was found to be: Bl(x ) x
24 SIMULATOR SIMULATION MAIN SYSTEM Notice that: As result of the simulation, it is possible to obtain mechanical answer of the loudspeaker mobile mass to an audio signal audio output deriving by the transduction. 24
25 SIMULATOR SUBSYSTEM SIMULATION REPRESENTING EQUATION: M ms x R x ms 1 C ms x F(x, Notice that: t) Input voltage generated by an audio signal normalized wave 25
26 SIMULATOR INPUT AND OUTPUT WAVEFORMS Comparison between input (yellow) an output (purple) waveforms Notice that: the instrument operates as a low-pass filter: the inertia of the mobile mass when high frequencies are present, the inductance of the coil causes a significant cut to the output wave amplitude. 26
27 CONCLUSIONS The analysis method, through a FEM software, of a common loudspeaker has been described. The study is based on the electromagnetic phenomena in the magnetic motor. Forces acting on the moving coil, magnetic energy stored and flux linkages have been studied in detail. Through the mechanical model it has been possible to study and observe the mechanical answer of the transducer to the input electromagnetic forces. An electromechanical simulator of the loudspeaker has been created linking these two analysis. Through the simulator it is possible to perform a first approximation study of the loudspeaker, that allows to design new devices or to improve existing models. This allows to limit the experimental tests and to verify the measurements on existing devices. 27
28 THANKS FOR YOUR ATTENTION! 22nd International Symposium on Power Electronics, Electrical Drives, Automation and Motion Ischia (Italy) June
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