Voice Coil Impedance as a Function of Frequency and Displacement Mark Dodd1, Wolfgang Klippel2, and Jack Oclee-Brown.3 1 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United Kingdom. [email protected] 2 Klippel GmbH, Dresden, 02177, Germany. [email protected] 3 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United Kingdom. [email protected] ABSTRACT Recent work by Klippel [1] and Voishvillo [2] has shown the significance of voice coil inductance in respect to the non-linear behaviour of loudspeakers. In such work the methods used to derive distortion require the inductance to be represented by an equivalent circuit rather than the frequency domain models of Wright and Leach. A new technique for measurement of displacement and frequency dependant impedance has been introduced. The complex relationship between coil impedance, frequency and displacement has been both measured and modelled, using FE, with exceptional agreement. Results show that the impedance model requires that its parameters vary independently with x to satisfactorily describe all cases. Distortion induced by the variation of impedance with coil displacement is predicted using both parameterised and FE methods, these predictions are compared to measurements of the actual distortion. The nature of this distortion is discussed. Dodd et al. Voice coil impedance also been predicted using FEM analysis. A new 1. INTRODUCTION impedance model with displacement varying parameters is discussed and fitted to the measured data. Finally, In order to describe the performance of a loudspeaker distortion induced by the variation of impedance with precisely, one must consider the electrical input coil displacement is predicted using both parameterised impedance at higher frequencies. The voice coil does and FE methods, these predictions are compared to not operate in free air but close to conducting and measurements of the actual distortion. The nature of this magnetically permeable structures: the pole tips, the distortion is discussed. magnet, the voice coil former, copper rings etc. The impedance can be only roughly modelled as a resistor 1.1. Glossary Re and an ideal inductance Le, the occurrence of eddy currents in these structures usually decreases the Bl(x) force factor (Bl product) inductance of the coil and increases losses at higher Cms(x) = 1 / Kms(x) mechanical frequencies [3]. compliance of loudspeaker suspension The nature of the voice coil impedance is dependent Exm exponent of imaginary part in WRIGHT upon the geometry and physical properties of the model magnet and surrounding assemblies. There are three Erm exponent of real part in WRIGHT prominent methods of deducing the coil (blocked) model impedance for a particular loudspeaker. Fm(x,i) reluctance force - The blocked impedance may be measured by Krm factor of real part in WRIGHT model clamping the voice coil stationary within the Kxm factor of imaginary part in WRIGHT magnet assembly to remove the motional part of the model impedance. Blocked impedance may then be K factor in LEACH model measured as with conventional impedance measurements. Kms(x) mechanical stiffness of loudspeaker suspension - Various impedance models have been developed in L (x) inductance used (cascaded model) order to describe the electrical behaviour of the e coil, these models may be fitted to conventional L2(x) inductance (cascaded model) unblocked impedance measurements. Leff(f,x) effective inductance depending on - Recently magnetic transient FEM has been Mms mechanical mass of loudspeaker successfully applied to deduce the blocked diaphragm assembly including air load impedance [4]. and voice coil n exponent in LEACH model At large amplitudes the impedance also varies significantly with voice coil displacement. Whereas the Rms mechanical resistance of total- frequency dependence of the impedance is a linear loudspeaker phenomenon effecting the amplitude response, but not Re electrical voice coil DC-resistance generating distortion, variation of impedance with R2(x) resistance (cascaded model) displacement generates significant harmonic and inter- R (f,x) effective losses due to eddy currents modulation distortion. eff depending on frequency and The nature of the impedance generated distortion is a displacement dependent upon the coil impedance, or more precisely Z (jω,x) complex excess impedance representing how the coil impedance varies with frequency & L the effect of the lossy inductance displacement of the coil. (motional impedance and Re is This paper investigates the voice coil impedance as a removed) function of frequency and of displacement. Measurements of frequency & displacement dependent ω angular frequency ω= 2πf impedance have been made using a quasi-static method using the LPM module of the Klippel analyser [5]. Frequency & displacement dependent impedance has Page 2 of 17 Dodd et al. Voice coil impedance 2. MODELLING electrical equivalent circuit (as shown in Figure 2 b) or as a digital IIR filter. 2.1. Lumped Parameter Model ZL(jω,x) a) Re Cms(jω)RMms ms ZL(jω) L2(x) v I Re Le(x) U Blv Bl Bli b) R2(x) Figure 1 Electro-mechanical equivalent circuit of the loudspeaker L (f, x) R (f, x) c) eff eff Figure 1 shows a simple equivalent circuit of a Figure 2 Representation of the electrical impedance loudspeaker system. The dominant nonlinearities are the variation of the force factor Bl(x), the mechanical 2.1.3. WRIGHT model compliance C (x) and the electrical impedance ms Wright [7] proposed a model using separate weighted ZL(jω,x) with voice coil displacement x. The DC resistance R and the motional impedance are not power functions in ω for both the real and imaginary e part of impedance. considered in the electrical impedance ZL(jω,x). Thus Erm Exm the impedance ZL(jω,x) may be measured at the ZL(jω)= Krm·ω + j·(Kxm·ω ) (3) electrical terminals by blocking the movement of the coil and subtracting the resistance measured at a very This model uses four free parameters and normally gives a low frequency. better fit than the other models with less parameters. Unfortunately, this function can not be directly realised as Different linear models have been developed to describe an analogue or digital system. the frequency dependency of ZL(jω,x) with a minimal number of free parameters: 2.1.4. Effective inductance Z (jω) = L ·(f)jω + R (f) 2.1.1. LEACH model L eff eff (4) M. Leach [6] proposed a weighted power function of the M. Leach also proposed normalising the imaginary part complex frequency as an approximation for ZL of the electrical impedance ZL(jω) to the frequency jω n ZL(jω)= K·(jω) ; ω= 2πf (1) and introducing an effective inductance Leff(f) which varies with frequency. The real part of ZL(jω) may be Although using only two free parameters this function considered as a frequency depending resistance Reff(f) can sometimes give a very good fit over a wide describing the losses due to eddy currents as shown in frequency range. Unfortunately, this function can not be Figure 2c . Though the number of parameters is very represented by an electrical equivalent circuit nor a high , two parameters for each frequency point, both simple digital system. parameters are easy to interpret and convenient for 2.1.2. LR-2 Model graphical representation. 2.1.5. Large signal modelling This model uses a series inductance Le connected to a second inductance L2 shunted by resistance R2. The linear models may be easily expanded to higher ZL(jω) = Le·jω + (R2·L2·jω ) / (R2 + L2·jω) (2) amplitudes by allowing each parameter to be dependent upon the displacement x. Although this model uses three free parameters it often For example considering the LR-2 model, the three provides a worse fit to measured ZL than the LEACH parameters Le(x), R2(x) and L2(x) are functions of the model. However, this model may be realised as an Page 3 of 17 Dodd et al. Voice coil impedance displacement x and may be approximated by a truncated Clearly the measurement of suspension stiffness using a power series expansion such as dc offset gives significantly different results from a dynamic measurement due to creep, relaxation and other N (5) i time dependent properties. Additionally, due to flux L ()xlx= ei∑ modulation, the force factor Bl(x) and the inductance i=0 Le(x) are dependent on the current i. Thus using an N (6) audio-like signal will produce far more meaningful data i R2 ()xrx= ∑ i than measurement with extremely small input current i=0 (coil offset generated by external force or pressure) or extremely large currents (coil offset generated by a dc N (7) i current). L2 ()xx= ∑λi i=0 Despite these limitations a quasi-static technique is a useful method for investigating the variation of the where the coefficients li, ri and λi are the free parameters of the model. impedance with both frequency and displacement. Measurements have been performed on two test 3. MEASUREMENT loudspeakers. Both loudspeakers share the following specifications: The application of a model to a particular real object Voice coil diameter: 2” nominal (51.30mm ID) usually requires an estimation of the free model Voice coil DCR: 6.72 Ohms parameters in such a way that the model describes the Turns in Voice coil: 126 Turns real object with maximal accuracy. Ferrite ring magnet With the linear models straightforward techniques are Annular low carbon steel top-plate available which may be applied for loudspeakers at Pole/plate assembly type yoke. small amplitudes. The magnet assembly of loudspeaker 1 has an Non-linear models require special techniques for the aluminium shorting ring placed above the magnetic gap. parameter identification. Static, quasi-static and full The magnetic assembly of loudspeaker 2 has an dynamic techniques have been developed to measure aluminium shorting ring placed below the magnetic gap.