Abstract:
An arrangement and method for converting an input signal z(t) into a mechanical or acoustical output signal p(t) comprising an electro-magnetic transducer using a coil at a fixed position and a moving armature, a sensor, a parameter measurement device and a controller. The parameter measurement device identifies parameter information P of an nonlinear model of the transducer considering and the saturation and the geometry of the magnetic elements. A diagnostic system reveals the physical causes of signal distortion and generates information for optimizing the design and manufacturing process of this transducer. The controller compensates for nonlinear signal distortion, stabilizes the rest position of the armature and protects the transducer against mechanical and thermal overload.

Description:
FIELD OF THE INVENTION 
       [0001]    The invention generally relates to an Arrangement and method for converting an input signal into an output signal and for generating a predefined transfer behavior between said input signal and said output signal. 
       BACKGROUND OF THE INVENTION 
       [0002]    The invention generally relates to an arrangement and a method for identifying the parameters of a nonlinear model of an electro-magnetic transducer and for using this information to correct the transfer characteristics of this transducer between input signal v and output signal p by changing the properties of the electro-magnetic transducer in design, manufacturing and by compensating actively undesired properties of said transducer by electric control. The electro-magnetic transducer may be used as an actuator (e.g. loudspeaker) or as a sensor (e.g. microphone) having an electrical input or output, respectively. 
         [0003]    Most loudspeakers, headphones and other electro-acoustical devices use an electro-dynamical transducer with a moving voice coil in a static magnetic field. Models have been developed for this kind of transducer which provide sufficient accuracy for measurement and control application, such as disclosed in U.S. Pat. No. 4,709,391, U.S. Pat. No. 5,438,625, U.S. Pat. No. 6,269,318, U.S. Pat. No. 5,523,715, DE 4336608, U.S. Pat. No. 5,528,695, U.S. Pat. No. 6,931,135, U.S. Pat. No. 7,372,966, U.S. Pat. No. 8,019,088, WO2011/076288A1, EP 1743504, EP 2453670, EP 2398253 and DE 10 2012 020 271. 
         [0004]    Electro-magnetic transducers converting an electric signal into a mechanic signal and vice versa use a coil at a fixed position and a moving armature connected via a driving pin with a diaphragm. This kind of transducer has some desired properties (e.g. high efficiency) which are not found in electro-dynamical transducers. The nonlinearities inherent in the electro-magnetic principle are a source of signal distortion. This disadvantage can be partly reduced by using a “balanced” armature using additional magnets. 
         [0005]    Straightforward distortion measurement techniques reveal harmonic distortion and other symptoms of nonlinearities inherent in this transducer. However, the results of these measurements do not give a complete description of the nonlinear transfer behavior but depend on the particular properties of the excitation stimulus. An accurate model of the electro-magnetic transducer is required to get a deeper insight in the physical causes and to predict the large signal performance for any input signal. The theory developed for electro-dynamical transducers is not applicable for electro-magnetic transducers. F. V. Hunt developed a first nonlinear model in “Electroacoustics—The Analysis of Transduction and Its Historical Background” (Acoustical Society of America, New York, 1954, 1982), which describes the electro-magnetic transducer by an electrical equivalent circuit comprising lumped elements. The inductance L(x), transduction factor T(x) and magnetic stiffness K mag (x) depend on the position x of the armature. This model was used by J. Jensen, et. al. in the paper “Nonlinear Time-Domain Modeling of Balanced-Armature Receivers,” published in the J. Audio Eng. Soc. Vol. 59, No. 3, 2011 March to predict the generation of odd-order harmonic distortion by assuming a symmetrical rest position of the armature in the magnetic field. All parameters are derived from the geometry of an ideal transducer having a magnetic material without saturation and hysteresis. The prior art has not disclosed a measurement technique for identifying the free parameters of this model applicable to real transducers. 
       SUMMARY OF THE INVENTION 
       [0006]    According to the present invention, the nonlinear model of the electro-magnetic transducer is extended to consider the saturation and hysteresis of the armature and other magnetic material. This extended model describes the dominant causes of nonlinear signal distortion in electro-magnetic transducers by using lumped parameters P such as coil inductance L(x,i), transduction factor T(x,i) and magnetic stiffness K mm (x,i) which are functions of the armature position x and current i. The nonlinear parameters correspond to a nonlinear flux function ƒ L (x,i) which describes the magnetic flux φ A  in the armature. 
         [0007]    The invention discloses a measurement technique which identifies all free parameters P of the extended model by monitoring at least one state variable of the transducer. The direct measurement of the armature position x or other mechanical or acoustical signals require a cost effective sensor. The hardware requirements can be reduced by monitoring an electrical signal at the terminals and by using the model for the identification of mechanical parameters. Optimal values of the free parameters P of the model are estimated by minimizing a cost function that describes the mean squared error between predicted and measured state variable. This measurement can be realized as an adaptive process while reproducing an arbitrary stimulus. The measurement is immune against ambient noise as found in a production environment or in the target application. The measurement technique evaluates the accuracy of the modeling by comparing the theoretical and real behavior of the transducer. 
         [0008]    The extended model with identified parameters reveals the physical causes of the signal distortion and their relationship to geometry, material of the components and problems caused by the assembling process in manufacturing. There are two alternative ways to use this information for correcting the vibration and the transfer behavior of the transducer: 
         [0009]    The parameters have a high diagnostic value for assessing design choices during the development process. The information is also useful for manufacturing and quality control. The offset x off =x s −x e , for example, is a meaningful characteristic for adjusting armature&#39;s equilibrium position x e . 
         [0010]    Active control using electric means and signal processing is an alternative way to compensate undesired effects of the transducer nonlinearities. A control law is derived from the results of the physical modeling. The free parameters of the control law correspond to the parameters P which are permanently identified by the adaptive measurement technique. No human expert is required to ensure the optimal control while properties of the transducer are varying over time due to aging, fatigue of the unit, climate, load changes as well as other external influences. 
         [0011]    The control system uses a state predictor to synthesize the states of the transducer under the condition that undesired nonlinear distortions are compensated in the output signal. This results in a control law with a feed-forward structure which is always stable. Any time delay may be added between the measurement system and the controller because the transferred parameter vector P changes slowly over time. The invention avoids any feedback of state variables from the measurement system to the controller. 
         [0012]    The control system can also be used for generating a DC component at the terminals of the transducer which moves the armature actively to the symmetry point x s  and reduces the offset x off  actively. This feature is very important for stabilizing transducers which have a low mechanical stiffness and which are desired for closed-box systems with a small intended leakage to cope with static air pressure variations. 
         [0013]    According to the invention the controller provides a protection against high amplitudes of the input signal causing a thermal and mechanical overload of the transducer which may cause excessive distortion in the output signal and a damage of the unit. The protection system uses the state vector x synthesized by a state predictor which corresponds to the state variables (e.g. armature position x, input current i) of the transducer to detect an overload situation. The limits of the permissible working range such as the maximal displacement x lim  may be automatically derived from the parameter vector P provided by the measurement system. 
         [0014]    These and other features, aspects and advantages of the present invention will become better understood with reference to the following drawings, description and claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]      FIG. 1  is a sectional view of a balanced-armature transducer. 
           [0016]      FIG. 2  shows a simplified magnetic circuit of the balanced-armature transducer. 
           [0017]      FIG. 3  shows a simplified model of the balanced-armature transducer using lumped parameters for modeling the electrical and mechanical components. 
           [0018]      FIG. 4  shows the electric input impedance of a balanced-armature transducer measured with a superimposed positive DC displacement X DC . 
           [0019]      FIG. 5  shows the electric input impedance of a balanced-armature transducer measured with a superimposed negative DC displacement x DC . 
           [0020]      FIG. 6  shows a magnetic circuit of the balanced-armature transducer according to the present invention. 
           [0021]      FIG. 7  shows an extended model of the balanced-armature transducer using lumped parameters for modeling the electrical and mechanical components according to the current invention. 
           [0022]      FIG. 8  shows a general identification and control system in accordance with the present invention. 
           [0023]      FIG. 9  shows an embodiment of the detector in accordance with the present invention. 
           [0024]      FIG. 10  shows the identified nonlinear inductance L(x,i=0) as a function of the position x of the armature. 
           [0025]      FIG. 11  shows the identified nonlinear inductance L(x e ,i) as a function of the input current i at the equilibrium point x e  of the armature. 
           [0026]      FIG. 12  shows an embodiment of the controller in accordance with the present invention. 
           [0027]      FIG. 13  shows an embodiment of the control law in accordance with the present invention. 
           [0028]      FIG. 14  shows an embodiment of the protection system in accordance with the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0029]    The derivation of the theory is illustrated by the example of the balanced-armature device as shown in  FIG. 1  but may be applied to other types of the electro-magnetic transducer in a similar way. The armature  1  is placed in the air gap between the magnets  3  and  5  which are part of a magnetic circuit  11 . A coil  7  placed at a fixed position generates a magneto-motive force Ni, depending on the number N of wire turns and input current i at the terminals  9 . The mechanical suspension  6  determines the rest position of the armature and the driving rod  10  is connected to the diaphragm  8 . 
         [0030]    The model, as disclosed by F. V. Hunt in the above mentioned prior art, is based on the assumptions that the magnets  3  and  5  have the same magneto-motive force 
         [0000]        F   m   =F   1   =F   2   (1)
 
         [0000]    and the magnetic reluctances R 1 (x) and R 2 (x) of the air in the upper and lower gap are much larger than any other reluctance in the iron path giving the simplified magnetic circuit in  FIG. 2 . Then the magnet fluxes φ 1  and φ 2  in the upper and lower gap, respectively, can be described by 
         [0000]    
       
         
           
             
               
                 
                   
                     Ni 
                     + 
                     
                       F 
                       m 
                     
                   
                   = 
                   
                     
                       φ 
                       1 
                     
                     
                       
                         ρ 
                         1 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     Ni 
                     - 
                     
                       F 
                       m 
                     
                   
                   = 
                   
                     
                       φ 
                       2 
                     
                     
                       
                         ρ 
                         2 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    using the non-linear permeances ρ 1 (x) and ρ 2 (x) which are the inverse of the reluctances R 1 (x) and R 2 (x), respectively. 
         [0031]    Assuming the armature  1  is symmetrically located at the initial rest position x=0 between the two demagnetized magnets, the resulting equilibrium point x e  corresponds to the symmetry point x s  after magnetizing the magnets. The permeances can be calculated by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ρ 
                       1 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           1 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             μ 
                             0 
                           
                            
                           A 
                         
                         
                           D 
                           - 
                           x 
                         
                       
                       = 
                       
                         
                           
                             
                               μ 
                               0 
                             
                              
                             A 
                           
                           
                             
                               D 
                               2 
                             
                             - 
                             
                               x 
                               2 
                             
                           
                         
                          
                         
                           ( 
                           
                             D 
                             + 
                             x 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       2 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           2 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             μ 
                             0 
                           
                            
                           A 
                         
                         
                           D 
                           + 
                           x 
                         
                       
                       = 
                       
                         
                           
                             
                               μ 
                               0 
                             
                              
                             A 
                           
                           
                             
                               D 
                               2 
                             
                             - 
                             
                               x 
                               2 
                             
                           
                         
                          
                         
                           ( 
                           
                             D 
                             - 
                             x 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    using the permeability μ 0  of air, cross section area A and the length D of the two air gaps for x=0. This modeling leads to the electrical equivalent circuit of the balanced-armature transducer as shown in  FIG. 3  comprising a transduction factor 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                             μ 
                             0 
                           
                            
                           
                             ANF 
                             m 
                           
                         
                         
                           D 
                           2 
                         
                       
                        
                       
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             D 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 x 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    an additional magnetic stiffness 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       K 
                       
                         m 
                          
                         
                             
                         
                          
                         m 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           F 
                           
                             m 
                              
                             
                                 
                             
                              
                             m 
                           
                         
                         x 
                       
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                             μ 
                             0 
                           
                            
                           
                             AF 
                             m 
                             2 
                           
                         
                         D 
                       
                        
                       
                         1 
                         
                           
                             ( 
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 x 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and a coil inductance 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       L 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             μ 
                             0 
                           
                            
                           
                             AN 
                             2 
                           
                         
                         D 
                       
                        
                       
                         
                           D 
                           2 
                         
                         
                           
                             D 
                             2 
                           
                           - 
                           
                             x 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    generating the reluctance force 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     rel 
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                      
                     
                       i 
                       2 
                     
                      
                     
                       
                         
                            
                           
                             L 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         
                            
                           x 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The magnetic stiffness K mm (x) is not found in the electro-dynamic transducer and is a unique feature of the electro-magnetic transducer. The moving mass M ms , the electrical DC resistance R e  of the coil and the mechanical resistance R ms  representing the losses in the mechanical system are linear parameters which are constant. 
         [0032]    Due to the denominator in Eq. (8) the inductance L(x) and the electrical input impedance Z e (f) at higher frequencies f increases for positive and negative displacement x. However, the results of practical measurement on real transducers reveal an impedance maximum at the equilibrium position x e  and a decrease of the impedance for positive and negative displacement as shown in  FIG. 4  and  FIG. 5 . Furthermore, the simple theory developed by F. V. Hunt neglects any offset of the initial position x=0 from the symmetry point x s  causing an asymmetry in the nonlinear parameter characteristics. 
         [0033]    Contrary to the prior art the reluctance R a (φ a )=ρ a (φ a ) −1  representing the armature in the magnetic circuit as shown in  FIG. 6  is a non-linear function depending on the magnetic flux φ a  corresponding to the fundamental equations 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     a 
                   
                   = 
                   
                     
                       φ 
                       1 
                     
                     - 
                     
                       φ 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     Ni 
                     + 
                     
                       F 
                       m 
                     
                   
                   = 
                   
                     
                       
                         φ 
                         1 
                       
                       
                         
                           ρ 
                           1 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         φ 
                         a 
                       
                       
                         
                           ρ 
                           a 
                         
                          
                         
                           ( 
                           
                             φ 
                             a 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     Ni 
                     - 
                     
                       F 
                       m 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           φ 
                           2 
                         
                         
                           
                             ρ 
                             2 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         φ 
                         a 
                       
                       
                         
                           ρ 
                           a 
                         
                          
                         
                           ( 
                           
                             φ 
                             a 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where x describes the absolute position of the armature. This position x=0 is determined by the mechanical suspension and describes the initial rest position of the armature with demagnetized magnets (F m =0) and no input current i=0. After magnetizing the magnets  3  and  5  and having a magneto-motive force (F m &gt;0) the armature is moved to an equilibrium position x e  where the magnetic DC force equals with the restoring force of the mechanical suspension. An input current i#0 generates a displacement x−x e  of the armature. 
         [0034]    The fluxes φ 1  and φ 2  in the upper and lower air gap, respectively, can be expressed by 
         [0000]    
       
         
           
             
               
                 
                   
                     φ 
                     1 
                   
                   = 
                   
                     
                       
                         
                           ρ 
                           1 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             F 
                             m 
                           
                           + 
                           Ni 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           
                             ρ 
                             1 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         
                           
                             ρ 
                             a 
                           
                            
                           
                             ( 
                             
                               φ 
                               a 
                             
                             ) 
                           
                         
                       
                        
                       
                         φ 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     2 
                   
                   = 
                   
                     
                       
                         
                           ρ 
                           2 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             F 
                             m 
                           
                           - 
                           Ni 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ρ 
                             2 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         
                           
                             ρ 
                             a 
                           
                            
                           
                             ( 
                             
                               φ 
                               a 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                           φ 
                           a 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0035]    The nonlinear functions of the permeances can be modeled by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ρ 
                             1 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           1 
                           
                             
                               R 
                               1 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               μ 
                               0 
                             
                              
                             A 
                           
                           
                             D 
                             - 
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   x 
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               D 
                               + 
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             ρ 
                             2 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           1 
                           
                             
                               R 
                               2 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               μ 
                               0 
                             
                              
                             A 
                           
                           
                             D 
                             + 
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   x 
                                   s 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               D 
                               - 
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with the symmetry point x s  describing the position x where the permeances of the upper and lower air gap are identical. 
         [0036]    According to Eq. (10) the flux in the armature can be calculated as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           a 
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 ρ 
                                 1 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   F 
                                   m 
                                 
                                 + 
                                 
                                   F 
                                   a 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 ρ 
                                 2 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   F 
                                   m 
                                 
                                 - 
                                 
                                   F 
                                   a 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 
                                   
                                     ρ 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     ρ 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   ρ 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     a 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               φ 
                               a 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               NDi 
                               + 
                               
                                 
                                   F 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   D 
                                   
                                     
                                       ρ 
                                       a 
                                     
                                      
                                     
                                       ( 
                                       
                                         φ 
                                         a 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   φ 
                                   a 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               f 
                               L 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 i 
                               
                               ) 
                             
                           
                            
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                             
                               D 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               NDi 
                               + 
                               
                                 
                                   F 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with the nonlinear flux function 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       L 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     1 
                     
                       1 
                       - 
                       
                         
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 s 
                               
                             
                             ) 
                           
                           2 
                         
                         
                           D 
                           2 
                         
                       
                       + 
                       
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               μ 
                               0 
                             
                              
                             A 
                           
                           D 
                         
                          
                         
                           1 
                           
                             
                               ρ 
                               a 
                             
                              
                             
                               ( 
                               
                                 
                                   φ 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     i 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    varying with the armature position x and the input current i. 
         [0037]    This function can be approximated by a series expansion 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       L 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     1 
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             
                               x 
                               - 
                               
                                 x 
                                 s 
                               
                             
                             D 
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           K 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               s 
                               k 
                             
                              
                             
                               ( 
                               
                                 i 
                                 + 
                                 
                                   
                                     s 
                                     x 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           s 
                                         
                                       
                                       D 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             2 
                              
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with the coefficients s k  describing the saturation of the magnetic material and the parameter s x  describing the dependency on armature position x. The first nonlinear term in the denominator represents the geometrical nonlinearity of the transducer and generates high values of f L (x,i) when x−x s  approaches ±D and the saturation is negligible (s k =0 for all k). The second term 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           μ 
                           0 
                         
                          
                         A 
                       
                       D 
                     
                      
                     
                       1 
                       
                         
                           ρ 
                           a 
                         
                          
                         
                           ( 
                           
                             φ 
                             a 
                           
                           ) 
                         
                       
                     
                   
                   &gt; 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           
                             x 
                             s 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       D 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    in the denominator representing the saturation becomes dominant in most transducers and the flux function decreases. If the parameter s x  is high, the saturation generated by position x of the armature may compensate the effect of the geometrical non-linearity in the first term of the denominator. 
         [0038]    The electrical mesh on the left-hand side of the equivalent circuit in  FIG. 7  corresponds to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         = 
                           
                          
                         
                           
                             
                               R 
                               e 
                             
                              
                             i 
                           
                           + 
                           
                             N 
                              
                             
                               
                                  
                                 
                                   φ 
                                   a 
                                 
                               
                               
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               R 
                               e 
                             
                              
                             i 
                           
                           + 
                           
                             N 
                              
                             
                               
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           μ 
                                           0 
                                         
                                          
                                         A 
                                       
                                       
                                         D 
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         f 
                                         L 
                                       
                                        
                                       
                                         ( 
                                         
                                           x 
                                           , 
                                           i 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         NDi 
                                         + 
                                         
                                           
                                             F 
                                             m 
                                           
                                            
                                           
                                             ( 
                                             
                                               x 
                                               - 
                                               
                                                 x 
                                                 s 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               R 
                               e 
                             
                              
                             i 
                           
                           + 
                           
                             
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                    
                                   i 
                                 
                                 ) 
                               
                             
                             
                                
                               t 
                             
                           
                           + 
                           
                             
                               T 
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   i 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                  
                                 x 
                               
                               
                                  
                                 t 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    comprising nonlinear inductance 
         [0000]    
       
         
           
             
               
                 
                   
                     L 
                      
                     
                       ( 
                       
                         x 
                         , 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       L 
                        
                       
                         ( 
                         
                           
                             x 
                             s 
                           
                           , 
                           0 
                         
                         ) 
                       
                     
                      
                     
                       
                         f 
                         L 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           i 
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     with 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 x 
                                 s 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 x 
                                 s 
                               
                               , 
                               
                                 i 
                                 = 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               μ 
                               0 
                             
                              
                             
                               AN 
                               2 
                             
                           
                           D 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the electro-magnetic transduction factor 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                        
                       
                         ( 
                         
                           x 
                           , 
                           i 
                         
                         ) 
                       
                     
                     ≈ 
                     
                       
                         T 
                          
                         
                           ( 
                           
                             
                               x 
                               s 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                        
                       
                         
                           f 
                           L 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             i 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         L 
                          
                         
                           ( 
                           
                             
                               x 
                               s 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                       λ 
                     
                      
                     
                       
                         f 
                         L 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           i 
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     with 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   λ 
                   = 
                   
                     
                       ND 
                       
                         F 
                         m 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0039]    The sum of the fluxes φ 1 +φ 2  in both air gaps can be expressed by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             φ 
                             1 
                           
                           + 
                           
                             φ 
                             2 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 ρ 
                                 1 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   F 
                                   m 
                                 
                                 + 
                                 Ni 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ρ 
                                 2 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   F 
                                   m 
                                 
                                 - 
                                 Ni 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   
                                     ρ 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     ρ 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 φ 
                                 a 
                               
                               
                                 
                                   ρ 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     a 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                             
                               
                                 D 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   Ni 
                                    
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   DF 
                                   m 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         s 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       φ 
                                       a 
                                     
                                     
                                       
                                         ρ 
                                         a 
                                       
                                        
                                       
                                         ( 
                                         
                                           φ 
                                           a 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0040]    Under the assumption that the saturation of the flux in the armature is the dominant nonlinearity in accordance with Eq. (20), the approximation 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       φ 
                       a 
                     
                     
                       
                         ρ 
                         a 
                       
                        
                       
                         ( 
                         
                           φ 
                           a 
                         
                         ) 
                       
                     
                   
                   ≈ 
                   
                     Ni 
                     + 
                     
                       
                         
                           F 
                           m 
                         
                          
                         
                           ( 
                           
                             x 
                             - 
                             
                               x 
                               s 
                             
                           
                           ) 
                         
                       
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and Eq. (26) gives the sum flux 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       φ 
                       1 
                     
                     + 
                     
                       φ 
                       2 
                     
                   
                   ≈ 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           μ 
                           0 
                         
                          
                         A 
                       
                       D 
                     
                      
                     
                       
                         F 
                         m 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0041]    The total driving force can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           F 
                           φ 
                         
                          
                           
                         = 
                         
                           
                             
                               
                                 φ 
                                 1 
                                 2 
                               
                               - 
                               
                                 φ 
                                 2 
                                 2 
                               
                             
                             
                               2 
                                
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   
                                     φ 
                                     1 
                                   
                                   + 
                                   
                                     φ 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 φ 
                                 a 
                               
                             
                             
                               2 
                                
                               
                                 μ 
                                 0 
                               
                                
                               A 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ≈ 
                           
                             
                               
                                 
                                   2 
                                    
                                   
                                     μ 
                                     0 
                                   
                                    
                                   
                                     AF 
                                     m 
                                     2 
                                   
                                 
                                 
                                   D 
                                   3 
                                 
                               
                                
                               
                                 
                                   f 
                                   L 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     i 
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   2 
                                    
                                   
                                     μ 
                                     0 
                                   
                                    
                                   
                                     ANF 
                                     m 
                                   
                                 
                                 
                                   D 
                                   2 
                                 
                               
                                
                               
                                 
                                   f 
                                   L 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     i 
                                   
                                   ) 
                                 
                               
                                
                               i 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             = 
                             
                               
                                 
                                   - 
                                   
                                     
                                       K 
                                       mm 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   T 
                                    
                                   
                                     ( 
                                     
                                       x 
                                       , 
                                       i 
                                     
                                     ) 
                                   
                                 
                                  
                                 i 
                               
                             
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    using the transduction factor T(x,i) according Eq. (24) and the magnetic stiffness 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       K 
                       mm 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           F 
                           mm 
                         
                         x 
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             
                               K 
                               mm 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   s 
                                 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                          
                         
                           
                             f 
                             L 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               i 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           - 
                           
                             
                               L 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     s 
                                   
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               λ 
                               2 
                             
                           
                         
                          
                         
                           
                             
                               f 
                               L 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 i 
                               
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0042]    The relationship between the forces in the mechanical system on the right-hand side of the equivalent circuit in  FIG. 7  can be described by 
         [0000]        T ( x,i ) i =( K ( x )− K (0)) x+K   mm ( x,i )( x−x   s )+ L   −1   [Z   m ( s ) s]*x,   (31)
 
         [0000]    using the inverse Laplace transformation L −1 [ ] and the convolution operator * to consider the mechanical impedance 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Z 
                         _ 
                       
                       m 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         K 
                          
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     + 
                     
                       R 
                       ms 
                     
                     + 
                     
                       
                         M 
                         ms 
                       
                        
                       s 
                     
                     + 
                     
                       
                         
                           Z 
                           _ 
                         
                         load 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    comprising the linear lumped parameters of the transducer and the impedance Z load (s) of the mechanic and acoustic load. 
         [0043]    The equilibrium point x e  of the armature can be found by 
         [0000]      ( K ( x   e )− K (0)) x   e   +K   mm ( x   e ,0)( x   e   −x   s )+ L   −1   [Z   m ( s ) s]*x   e =0  (33)
 
         [0000]    using Eq. (31) with input current i=0. 
         [0044]    Contrary to the prior art the nonlinear inductance L(x,i), transduction factor T(x,i) and magnetic stiffness K mm (x,i) are nonlinear functions of displacement x and current i. The differential equations of the balanced-armature transducer can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   u 
                   = 
                   
                     
                       
                         R 
                         e 
                       
                        
                       i 
                     
                     + 
                     
                       
                         L 
                          
                         
                           ( 
                           
                             
                               x 
                               s 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                        
                       
                         
                            
                           
                             ( 
                             
                               i 
                               · 
                               
                                 
                                   f 
                                   L 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         
                            
                           t 
                         
                       
                     
                     + 
                     
                       
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 x 
                                 s 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                         λ 
                       
                        
                       
                         
                           f 
                           L 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             i 
                           
                           ) 
                         
                       
                        
                       
                         
                            
                           x 
                         
                         
                            
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                           e 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               s 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                       λ 
                     
                      
                     
                       
                         f 
                         L 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           i 
                         
                         ) 
                       
                     
                      
                     i 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             K 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                           - 
                           
                             K 
                              
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         ) 
                       
                        
                       x 
                     
                     - 
                     
                       
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 x 
                                 s 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                         
                           λ 
                           2 
                         
                       
                        
                       
                         
                           f 
                           L 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             i 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           - 
                           
                             x 
                             s 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           L 
                           
                             - 
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   Z 
                                   _ 
                                 
                                 m 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             s 
                           
                           ] 
                         
                       
                       * 
                       
                         x 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0045]    After developing the stiffness K(x) of the mechanical suspension into a power series by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       K 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         K 
                       
                        
                       
                           
                       
                        
                       
                         
                           k 
                           k 
                         
                          
                         
                           x 
                           k 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    the free parameters of the model 
         [0000]        P=[P   1    . . . P   j    . . . P   j ] T   =[P   lin   P   nlin   ]=[P   lin   P   mag   P   sus ]  (37)
 
         [0000]      comprise a linear parameter vector 
         [0000]        P   lin   =└R   e   M   ms   L ( x   off ,0) R   ms   λk   0 ┘(38)
 
         [0000]      and a nonlinear parameter vector  P   nlin  which can be separated into parameters of the magnetic circuit 
         [0000]        P   mag   =└x   off   s   x   Ds   1    . . . s   K   (39)
 
         [0000]    and parameters of the mechanic or acoustic suspension 
         [0000]        P   sus   =[k   1    . . . k   K ].  (40)
 
         [0046]    The nonlinear mechanical parameters P sus  of the suspension are also found in an electro-dynamical loudspeaker. The nonlinear magnetic parameters P nlin  are different from the inductance L(x,i) and the force factor Bl(x) found in a moving-coil transducer where the two parameters have a completely different curve shape. In a balanced-armature transducer the flux function ƒ L (x,i) generates a similar nonlinear curve shape of the inductance L(x,i), transduction factor T(x,i) and magnetic stiffness K mm (x,i). The magnetic stiffness K mm (x,i) generated in the magnetized transducer does not exist in electro-dynamical transducers. 
         [0047]    The extended model of the electro-magnetic transducer is the basis for the arrangement  30  shown in  FIG. 8 . The balanced-armature transducer  25  is operated in a closed box system  14  where the enclosure has a defined leakage  16 . The input current i and voltage u at the terminals of the transducer are measured by using a sensor  13  and are supplied to the inputs  17  and  19  of a parameter measurement system  15  generating the optimal parameter vector P at the measurement output  23 . The parameter vector P is supplied to the parameter input  21  of the controller  29  as well as to the input of a diagnostic system  22  generating diagnostic information (e.g. offset x off  of the armature). The controller receives the input signal v at the control input  31  and generates the control output signal u transferred via the DA-converter  27  and a power amplifier  63  to the transducer  25 . 
         [0048]    According to the invention an optimal estimate of the parameter vector P is determined in the measurement system  15  as shown in  FIG. 9  by calculating the error signal 
         [0000]        e=û−u   (41)
 
         [0000]    in the model evaluation system  71  as the difference between the voltage a predicted by the nonlinear model  73  and measured voltage u. 
         [0049]    Two parameter estimators  80 ,  84  determine optimal parameters P lin , P nlin  in vector P by searching for the minimum of the mean squared error 
         [0000]        C=MSE=E{e ( t ) 2 }.  (42)
 
         [0000]    This objective can be accomplished by the LMS-algorithm 
         [0000]        P[n]=P[n− 1 ]+μe ( t ) G ( t )  (43)
 
         [0000]    realized by systems  75 ,  79  with the step size μ and the gradient vector 
         [0000]    
       
         
           
             
               
                 
                   
                     G 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           G 
                           lin 
                         
                          
                         
                             
                         
                          
                         
                           G 
                           min 
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               ∂ 
                               u 
                             
                             
                               ∂ 
                               
                                 P 
                                 1 
                               
                             
                           
                            
                           … 
                            
                           
                             
                               ∂ 
                               u 
                             
                             
                               ∂ 
                               
                                 P 
                                 j 
                               
                             
                           
                            
                           … 
                            
                           
                             
                               ∂ 
                               u 
                             
                             
                               ∂ 
                               
                                 P 
                                 J 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    generated in the gradient systems  81 ,  85  by using input current i. 
         [0050]    The nonlinear model  73  comprises a first subsystem  91  generating the voltage û in accordance with Eq. (34) and provides this value to the non-inverting input of the model evaluation system  71 . A second subsystem  89  generates the position 
         [0000]    
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 L 
                                 e 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     s 
                                   
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                             λ 
                           
                            
                           
                             
                               f 
                               L 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 i 
                               
                               ) 
                             
                           
                            
                           i 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 K 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               - 
                               
                                 K 
                                  
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             ) 
                           
                            
                           x 
                         
                         + 
                         
                           
                             
                               L 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     s 
                                   
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               λ 
                               2 
                             
                           
                            
                           
                             
                               f 
                               L 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 i 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 s 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                     * 
                     
                       
                         L 
                         
                           - 
                           1 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           
                             
                               
                                 
                                   Z 
                                   _ 
                                 
                                 m 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             s 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    in accordance with Eq. (35) and supplies this signal to subsystems  87 ,  91 . The third subsystem  87  generates the instantaneous value of the flux function ƒ L (x,i) in accordance with Eq. (19) using the parameter P mag  and supplies this value to the subsystems  89  and  91 . The measured current i is the input of the subsystems  87  and  89 . 
         [0051]      FIG. 10  shows the nonlinear inductance L(i=0,x−x e ) versus displacement x−x e  from the equilibrium position x e  with input current i=0 calculated by using parameters P mag . The position at maximum inductance corresponds to the symmetry point x s . The decay of the inductance for larger displacements agrees with the decrease of the electrical input impedance at higher frequencies as shown in  FIG. 4  and  FIG. 5 .  FIG. 11  and shows the dependency of the inductance L(i, x e ) versus input current i at the equilibrium point x e . 
         [0052]    According to the invention a diagnostic system  22  derives information from the identified parameter vector P which is the basis for improving the electro-magnetic transducer during development and manufacturing. The symmetry point x s  in vector P mag  reveals the optimal rest position of the armature and the offset x off =x s −x e  to the equilibrium position x e . If the magnets  3 ,  5  have not been magnetized and the armature is at the initial rest position x=0 the sign and the amount of x s  can be used to adjust the rest position of the mechanical suspension in one step. After adjusting the initial rest point x=0 of the armature to the symmetry point x s =0 the equilibrium position x e =0 with magnetized magnets will also stay at the initial rest point (if the transducer behaves stable). 
         [0053]    Bifurcation and other unstable behavior can be avoided by ensuring the condition 
         [0000]      − K   mm ( x, 0)( x−x   s )&lt;( K ( x )− K (0)) x+L   −1   [Z   m ( s ) s]*x.   (46)
 
         [0000]    This condition can be realized by generating dominant saturation in the magnetic circuit according to Eq. (20) and/or sufficient restoring force of the mechanical suspension. The nonlinear stiffness variation in K(x)−K(0) of the suspension revealed by the coefficients k j  in P sus  can be used to stabilize the transducer and to generate a desired transfer characteristic. The parameters s k  in vector P mag  reveal the dominant nonlinearity in the denominator of Eq. (19) and parameter s x  shows which state variable (current i or position x) has the largest influence on this process. This information can be used to find the optimal cross section area A a  of the armature  1  where the nonlinear saturation compensates the effect of the geometrical nonlinearity. 
         [0054]    According to a further objective of the invention the identified parameter vector P is also used to compensate actively undesired nonlinearities of the electro-magnetic transducer by using an electric controller  29  and generating a desired transfer behavior of the overall system (controller  29 +transducer  25 ). 
         [0055]      FIG. 12  shows an embodiment of the controller in accordance with the invention. The input signal v at input  31  is supplied via a protection system  42  to the input  43  of the control law system  39  generating the control output signal u at control output  49 . The controller also contains a state predictor  37  generating the state vector x which comprises position x, current i and other state variables of the transducer. 
         [0056]    The linearization of the armature movement will also give a linear acoustical output of the transducer while assuming that the sound radiation by the diaphragm  8  is a linear process. Thus, the following linear relationship 
         [0000]    
       
         
           
             
               
                 
                   
                     x 
                     = 
                     
                       ( 
                       
                         w 
                         - 
                         
                           
                              
                             
                               ( 
                               
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         s 
                                       
                                       , 
                                       0 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   i 
                                   l 
                                 
                               
                               ) 
                             
                           
                           
                              
                             t 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         L 
                           
                         
                           - 
                           1 
                         
                           
                           
                         * 
                       
                        
                       
                         { 
                         
                           
                             T 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   s 
                                 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   
                                     R 
                                     e 
                                   
                                    
                                   
                                     
                                       Z 
                                       m 
                                     
                                      
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     T 
                                      
                                     
                                       ( 
                                       
                                         
                                           x 
                                           s 
                                         
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               ) 
                             
                              
                             s 
                           
                         
                         } 
                       
                     
                     + 
                     
                       x 
                       s 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0000]    between controller input signal w input and position x of the armature requires a particular nonlinear transfer characteristic of the control law system  39  defined by 
         [0000]        u =α( x )[ w +β( x )]  (48)
 
         [0000]    with the control gain 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         T 
                          
                         
                           ( 
                           
                             
                               x 
                               s 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                       
                         T 
                          
                         
                           ( 
                           
                             x 
                             , 
                             i 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       1 
                       
                         
                           f 
                           L 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the control additive 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           β 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       T 
                                        
                                       
                                         ( 
                                         
                                           x 
                                           , 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                   
                                     
                                       T 
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             s 
                                           
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     2 
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             
                               T 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     off 
                                   
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                              
                             v 
                           
                           - 
                           
                             
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         
                                           x 
                                           s 
                                         
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     i 
                                     l 
                                   
                                 
                                 ) 
                               
                             
                             
                                
                               t 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 R 
                                 e 
                               
                               
                                 T 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       s 
                                     
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         K 
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         K 
                                          
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   x 
                                 
                                 + 
                                 
                                   
                                     
                                       K 
                                       mm 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               T 
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   i 
                                 
                                 ) 
                               
                             
                             
                               T 
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                    
                                   i 
                                 
                                 ) 
                               
                             
                             
                                
                               t 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       F 
                                       l 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       s 
                                     
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                               λ 
                             
                              
                             v 
                           
                           - 
                           
                             
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         
                                           x 
                                           s 
                                         
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     i 
                                     l 
                                   
                                 
                                 ) 
                               
                             
                             
                                
                               t 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 
                                   R 
                                   e 
                                 
                                  
                                 λ 
                               
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       s 
                                     
                                     , 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         K 
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         K 
                                          
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   x 
                                 
                                 - 
                                 
                                   
                                     
                                       l 
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             s 
                                           
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     
                                       λ 
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       f 
                                       L 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         i 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               f 
                               L 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 i 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             s 
                                           
                                           , 
                                           0 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         f 
                                         L 
                                       
                                        
                                       
                                         ( 
                                         
                                           x 
                                           , 
                                           i 
                                         
                                         ) 
                                       
                                     
                                      
                                     i 
                                   
                                   ) 
                                 
                               
                               
                                  
                                 t 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0057]      FIG. 13  shows an embodiment of the control law system  39  comprising an adder  51  and a multiplier  65  in accordance with Eq. (48), an additive sub-controller  60  in accordance with Eq. (50) and a multiplicative sub-controller  61  in accordance with Eq. (49). A nonlinear subsystem  59  identical with the second subsystem  89  is provided with the nonlinear parameter P mag  from input  47  and with the armature position x and current i from the state vector input  45  and generates the instantaneous value of the flux function ƒ L (x,i) supplied to the transfer systems  57 ,  55  and  53 . The instantaneous inductance L(x,i) generated in  57  in accordance with Eq. (22) and the magnetic stiffness K mm (x,i) in  55  in accordance with Eq. (30) is supplied to the additive sub-controller  60 . The transduction factor T(x,i) generated in  53  in accordance with Eq. (24) is supplied to both sub-controllers  60  and  61 . 
         [0058]    The state vector x=[x,v,i l ,i] T  generated in state expander  37  also comprises the velocity 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     = 
                     
                       
                          
                         x 
                       
                       
                          
                         t 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    the linear current i l  generated by 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     l 
                   
                   = 
                   
                     
                       
                         L 
                         
                           - 
                           1 
                         
                       
                        
                       
                         { 
                         
                           
                             
                               
                                 
                                   Z 
                                   _ 
                                 
                                 m 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             s 
                           
                           
                             T 
                             
                               ( 
                               
                                 
                                   x 
                                   s 
                                 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                         
                         } 
                       
                       * 
                       x 
                     
                     = 
                     
                       
                         L 
                         
                           - 
                           1 
                         
                       
                        
                       
                         { 
                         
                           λ 
                            
                           
                             
                               
                                 
                                   
                                     Z 
                                     _ 
                                   
                                   m 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                                
                               s 
                             
                             
                               L 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     s 
                                   
                                   , 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                       * 
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the predicted nonlinear current generated by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         i 
                         = 
                         
                           
                             
                               T 
                                
                               
                                 ( 
                                 
                                   
                                     x 
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         [0059]    The controller  29  also compensates for the offset x off  actively and ensures that the equilibrium point x e  is identical with the symmetry point x s  of the magnetic circuit. This requires that the power amplifier  27  is DC-coupled to transfer the DC component generated in the controller  29  to the transducer  25 . This ensures maximum excursion generated by the external stimulus w and a symmetrical limiting of armature at the upper and lower pole tips. 
         [0060]    An unstable transducer as defined by Eq. (46) can also be stabilized by active control when the symmetry point x s  is permanently updated using a high step size parameter μ in Eq. (43) to realize a short measurement time T m . The step size parameter can be reduced if the electro-magnetic transducer  25  is operated in a sealed enclosure  14  having a small air leak  16  required to compensate for variation of the static air pressure. The additional stiffness of the enclosed air stabilizes the equilibrium point for a short time τ B  required by the air to pass the leak. If the measurement time T m  is shorter than the time τ B  the active control can compensate any offset x off =x s −x e  or instability of the armature. This technique makes it possible to reduce the stiffness K(x) of the mechanical suspension and to increase the acoustical output of the transducer in a closed box  14  at low frequencies. 
         [0061]    According to the third objective of the invention the identified parameter vector P is also used to protect the electro-magnetic transducer against mechanical and thermal overload. The embodiment of the protection system  42  shown in  FIG. 12  comprises a protection control system  35 , an attenuator  40  connected in series to a high-pass filter  41 . A control signal C T  provided from the output  102  of the protection control system  35  attenuates all spectral components in signal w in the case of thermal overload. The control signal C x  from the output  103  increases the cut-off frequency of the high-pass filter  41  and attenuates the low frequency components in the case of mechanical overload. 
         [0062]      FIG. 14  shows an embodiment of the protection control system  35  which receives the state vector x at input  104  and the parameter vector P at input  101 . The nonlinear modeling of the electrical circuit in Eq. (34) ensures an accurate estimation of the DC resistance R e (T c ) in the vector P lin  which is a function of the instantaneous coil temperature T c . Comparing the instantaneous value of R e (t) with the initial value R e (t=0) in the thermal control subsystem  115  reveals the increase of the coil temperature ΔT=T c (t)−T c (t=0). If the increase of the coil temperature exceeds a permissible limit value ΔT lim  the control signal C T  attenuates the input signal v to prevent a thermal overload. 
         [0063]    The instantaneous position x(t) of the armature generated in the state estimator  37  of the controller can also be used for providing a protection of the armature  1 , suspension  6 , driving pin  10 , diaphragm  8  and other mechanical elements of the transducer. If the absolute value of the armature displacement └x(t)−x e ┘ exceeds a permissible displacement limit Δx lim  the mechanical control subsystem  117  activates the control signal C x . The displacement limit Δx lim  is determined by a working range detector  125  receiving the parameter vector P. The working range detector  125  comprises a minimum detector  113 , a mechanical detector  119  and a magnetic detector  121 . 
         [0064]    The minimum detector  113  searching for the minimal value between limit x mag  generated by a magnetic detector  121  and a limit x sus  generated by a mechanical detector  119 . 
         [0065]    The magnetic detector  121  receives the parameters P mag  and generates two sub-limits: The first sub-limit x sat  is generated by system  105  using the nonlinear flux function ƒ L (x,i) generated by nonlinear system  107  in accordance with Eq. (19) and searching for the displacement where the value of f L (x sat ,i=0)=T sat  equals a permissible threshold T sat . The second sub-limit x D  is determined by system  113  which corresponds to parameter D in parameter vector P mag  indicating the displacement where the armature hits the upper or lower pole tip. The minimum of x D  and x sat  gives the limit x mag . 
         [0066]    The mechanical detector  119  receives the parameters P sus  and generates the relative stiffness function K(0)/K(x) of the suspension  6  in the nonlinear system  111  using Eq. (36). The solver  109  searches for the limit x sus  where the variation of the nonlinear stiffness K(0)/K(x sus )=T sus  equals a permissible threshold T sus .