Abstract:
The invention relates to an improved electro-dynamic loudspeaker. The electro-dynamic loudspeaker comprises (a) a voice coil for generating an acoustic waveform, the voice coil being longitudinally movable from an initial rest position to generate the acoustic waveform; (b) a second element of the loudspeaker, the second element being stationary relative to the voice coil; (c) an inductance-affecting core mounted on the voice coil for movement therewith, the inductance-affecting core having a length and a variable inductance-affecting capacity; (d) at least one inductor adjoining the inductance-affecting core and mounted on the second element, the at least one inductor having an associated length shorter than the length of the conductor core such that only a variable portion of the inductance-affecting core adjoins the inductor, the variable portion having a variable average inductance-affecting capacity and a portion length substantially equal to the associated length of the at least one inductor; and, (e) a position sensor circuit connected to the at least one inductor for providing a variable signal based on the variable average inductance-affecting capacity of the variable portion of the inductance-affecting core adjoining the at least one inductor. The variable average inductance-affecting capacity of the variable portion varies with the degree of deflection of the voice coil relative to the second element to vary the variable signal.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates to a position sensor. More particularly, it relates to a position sensor for providing an electrical signal that varies in a selected manner with the placement of a voice coil from an at rest position, and a method of constructing same.  
         BACKGROUND OF THE INVENTION  
         [0002]    The construction and operation of electro-dynamic loudspeakers are well known. The physical limitations in their construction are one cause of non-linear distortion, which is sensible in the generated sound production. Distortion is particularly high at low frequencies, in relatively small sealed box constructions where cone displacement or excursions are at their maximum limit.  
           [0003]    In the past, one of many approaches taken to reduce speaker distortion has been to use motional feedback to compensate for this distortion. Motional feedback controls frequency response and reduces non-linear distortions. Motional feedback is usually implemented using accelerometers, velocity sensors and/or position sensors. In the past, accelerometers have been the most successful, as they are inexpensive and their performance does not depend on the extent of displacement, thereby contributing to the linearity of the output signal. The linearity of any sensor is critical in audio applications, as even very strong feedback cannot reduce distortions beyond those introduced by the sensor itself.  
           [0004]    Despite the advantages afforded by the linearity of their output, accelerometers have problems of their own. At low frequencies, the distortions generated by typical speakers are very high. Some components of these distortions can move the speaker cone from its optimal, center position; however, accelerometers will be blind to slow shift in cone position and their output signals will not include information that can be sent back to the amplifier to correct for this slow shift. Similarly, velocity sensors will be blind to cone position.  
           [0005]    Position sensors do not suffer from these shortcomings. However, like velocity centers, the operation of position sensors requires two elements to be moved relative to each other. This makes their operation sensitive to cone excursion. Consequently, the signals provided by each will not be linear, particularly at large displacements  
           [0006]    Thus, there is a need to measure slow shift and cone position. Both accelerometers and velocity sensors are unable to provide this measurement. Position sensors can provide this measurement; however, such sensors themselves create non-linearities. Position sensors that measure the variations in coil induction are generally considered to be the most practical, reliable and least sensitive to the environment of available position sensors. However, such position sensors still suffer from these problems. Existing sensors of this kind typically include multiple coils mounted coaxially with a voice coil of a speaker. A conductive element such as a metal rod or another coil moves inside the external coils. An electrical circuit converts the movement of the interior conductive element in the exterior coil to an electrical signal. However, as described above, the conversion of the displacement to voltage may not be linear, especially for large displacements. In addition, as the coils are mounted coaxially with the speaker voice coil, additional voltages may be induced in the voice coils thereby generating noise.  
           [0007]    Accordingly, there is a need for a position sensor that is inexpensive, easy to build, provides a linear output and minimizes the generation of voltage noise in the speaker voice coil.  
         SUMMARY OF THE INVENTION  
         [0008]    An object of an aspect of the present invention is to provide an improved position sensor.  
           [0009]    In accordance with this aspect of the present invention there is provided a position sensor for measuring a degree of deflection of a first element relative to a second element. The position sensor comprises (a) an inductance-affecting core mounted on the first element for movement therewith, the inductance-affecting core having a length and a variable inductance-affecting capacity varying along the length; (b) at least one inductor adjoining the inductance-affecting core and mounted on the second element, the at least one inductor having an associated length shorter than the length of the conductor core such that only a variable portion of the inductance-affecting core adjoins the inductor, the variable portion having a variable average inductance-affecting capacity and a portion length substantially equal to the associated length of the at least one inductor; and, (c) a position sensor circuit connected to the at least one inductor for providing a variable signal based on the variable average inductance-affecting capacity of the variable portion of the inductance-affecting core adjoining the at least one inductor. The variable average inductance-affecting capacity of the variable portion varies with the degree of deflection of the first element relative to the second element to vary the variable signal.  
           [0010]    An object of a second aspect of the present invention is to provide a method of designing a position sensor for providing an output that varies linearly with displacement.  
           [0011]    In accordance with the second aspect of the present invention, there is provided a method of measuring a degree of deflection of a first element relative to a second element. The method comprises (a) selecting a selected variable output signal for measuring the degree of deflection, wherein the variable output signal varies with the degree of deflection; (b) mounting an inductance-affecting core on the first element for movement therewith, the inductance-affecting core having a length and a variable inductance-affecting capacity; (c) mounting at least one inductor on the second element adjoining the inductance-affecting core, the at least one inductor having an associated length shorter than the length of the conductor core such that only a variable portion of the inductance-affecting core adjoins the inductor, the variable portion having a variable average inductance-affecting capacity; (d) connecting the at least one inductor to a position sensor circuit for providing the selected variable output signal based on the variable average width of the variable portion of the position sensor; and (e) configuring the inductance-affecting core to have the variable inductance-affecting capacity required to provide the selected variable signal.  
           [0012]    An object of a third aspect of the present invention is to provide an improved loudspeaker.  
           [0013]    In accordance with the third aspect of the present invention, there is provided an electro-dynamic loudspeaker. The electro-dynamic loudspeaker comprises (a) a voice coil for generating an acoustic waveform, the voice coil being longitudinally movable from an initial rest position to generate the acoustic waveform, (b) a second element of the loudspeaker, the second element being stationary relative to the voice coil; (c) an inductance-affecting core mounted on the voice coil for movement therewith, the inductance-affecting core having a length and a variable inductance-affecting capacity; (d) at least one inductor adjoining the inductance-affecting core and mounted on the second element, the at least one inductor having an associated length shorter than the length of the conductor core such that only a variable portion of the inductance-affecting core adjoins the inductor, the variable portion having a variable average inductance-affecting capacity and a portion length substantially equal to the associated length of the at least one inductor, and, (e) a position sensor circuit connected to the at least one inductor for providing a variable signal based on the variable average inductance-affecting capacity of the variable portion of the inductance-affecting core adjoining the at least one inductor. The variable average inductance-affecting capacity of the variable portion varies with the degree of deflection of the voice coil relative to the second element to vary the variable signal  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]    For a better understanding of the present invention and to show more clearly how it may be carried into effect, reference will now be made, by way of example, to the accompanying drawings, which show preferred embodiments of the present invention, and in which:  
         [0015]    [0015]FIG. 1 is a perspective side view of a first embodiment of a position sensor in accordance with the present invention;  
         [0016]    [0016]FIG. 2 illustrates, in a perspective side view, an alternative embodiment of the position sensor shown in FIG. 1,  
         [0017]    [0017]FIG. 3 illustrates, in a schematic diagram, an electrical sensor circuit used in combination with the position sensor of FIG. 2 in a further embodiment of the invention;  
         [0018]    [0018]FIG. 4, in a sectional view, illustrates a cross section of the mechanical construction of the speaker device and the relative position of the position sensor;  
         [0019]    [0019]FIG. 5 is a graph plotting the output voltage produced by a prior art position sensor against the displacement of a triangular conductive core of the position sensor;  
         [0020]    [0020]FIG. 6 is a graph plotting the width of the conductive core of FIG. 5 against its displacement;  
         [0021]    [0021]FIG. 7 is a graph plotting the width of a conductive core of a position sensor of FIG. 5 against the output voltage of the position sensor;  
         [0022]    [0022]FIG. 8 is a graph plotting width of a conductive core of the linear position sensor in accordance with a further embodiment of the invention against a displacement of the conductive core;  
         [0023]    [0023]FIG. 9 is a graph plotting the output voltage produced by the linear position sensor of FIG. 8 against the displacement of the linear position sensor;  
         [0024]    [0024]FIG. 10 is a graph plotting the ratio of the force factor at a particular displacement of a voice coil to the force factor at a rest position against the displacement of the voice coil;  
         [0025]    [0025]FIG. 11 is a graph plotting width of a conductive core of an inverse parabolic position sensor in accordance with a further embodiment of the invention against a displacement of the conductive core;  
         [0026]    [0026]FIG. 12 is a graph plotting the output voltage produced by the inverse parabolic position sensor of FIG. 11 against the displacement of this inverse parabolic position;  
         [0027]    [0027]FIG. 13 is a graph plotting width of a conductive core of a parabolic position sensor against displacement of the conductive core;  
         [0028]    [0028]FIG. 14 is a graph plotting the output voltage produced by the parabolic position sensor of FIG. 13 against the displacement of a parabolic position sensor; and,  
         [0029]    [0029]FIG. 15 is a schematic diagram of a loudspeaker with a motional feedback system for reducing non-linear distortion of the loudspeaker. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0030]    [0030]FIG. 1 illustrates a position sensor device  20 , which includes a first and second inductance coil  22 ,  24  and an approximately triangular shaped conductive core  26 . Optionally, all of these components  22 ,  24 , are manufactured on printed circuit boards (PCB). Furthermore, the coils may be printed on both sides of the PCB boards and electrically connected in series in order to maximize their total inductance A conductive region  28  of the conductive core  26  is longitudinally displaced within a finite gap region, defined by  30 . As the conductive core  26  moves in the direction indicated by Arrow X, a larger amount of copper is immersed in the magnetic field generated by the coils  22 ,  24  This in turn decreases the inductance of the coils  22 ,  24 . Conversely, as the conductive core  26  moves in a direction indicated by Arrow Y, a smaller amount of copper is immersed in the magnetic field generated by the coils  22 ,  24 , which in turn increases the inductance of the coils  22 ,  24 . The conductive core  26  is geometrically compensated in order to ensure that its longitudinal displacement (X or Y Arrow direction) in the center of the finite gap region  30  generates a linear change in the output voltage of the position sensor circuit. Hence, a linear position control signal (position sensor output shown in FIG. 3) is generated as a result of this inductance change. As illustrated in FIG. 1, the shape of the conducting region  28  is not precisely triangular. It is shaped to linearize the relationship between the output voltage of the position sensor  20  and the displacement of the core  26 . Conducting region  28  has a curved shape. As illustrated in FIG. 1, in use, the first and second inductance coils  22 ,  24  are stationary, whilst the conductive core  26  is attached to a bobbin  32  (FIG. 4) of a voice coil  34 . Therefore, as the voice coil  34  longitudinally moves, the conductive core  26  is longitudinally displaced within the finite gap region  30  between the coils  22 ,  24 . Hence, the inductance of the coils  22 ,  24  varies in unison with voice coil movement. Although the coils  22 ,  24  are stationary and the conductive core  26  moves, in an alternative embodiment, it will be appreciated that the coils  22 ,  24  may be connected to the voice coil  34 , whilst the conductive core  26  remains stationary. However, it is found that by connecting the core  26  to the voice coil  34 , a rigid connection which generates satisfactory position sensing is provided.  
         [0031]    [0031]FIG. 2 shows an alternative embodiment of the position sensor  20 , wherein the conductive core  26  is comprised solely of a conductive region. The operation of this sensor is essentially the same as that of the sensor described and illustrated in FIG. 1.  
         [0032]    Referring to FIG. 1, the position sensor  20  is also positioned, such that no electrical cross talk occurs between the inductance coils  22 ,  24  and the voice coil  34 . This is achieved ensuring that the vector orientation of the magnetic field generated by the inductance coils  22 ,  24  is orthogonal to the vector orientation of the magnetic field generated by the voice coil  34 . In terms of the physical positioning of the inductance coils  22 ,  24  and the voice coil  34 , their respective axes must be orthogonal in order to eliminate electrical cross talk. This means that a concentric longitudinal axis  36 , which passes concentrically through the voice coil  34  must be orthogonal to a first axis  38  which passes through the center of both inductance coils  22 ,  24   
         [0033]    [0033]FIG. 3 illustrates the position sensor circuit comprising the position sensor device  20  and processing circuit  46 . The circuit  46  converts the changes in the inductance of the position sensor  20  and generates the position control signal  48  wherein the voltage magnitude of the position control signal  48  is proportional to the displacement of the core  26  Within the circuit of FIG. 3, an oscillator circuit  50  comprises a crystal (6 MHz, for example)  52 , capacitor component  54 , capacitor component  56 , resistor component  58 , resistor component  60 , XOR logic gate  62  and XOR logic gate  64 . This circuit  50  generates a 6 MHz squarewave signal at the output  66  of XOR gate  64 . The 6 MHz squarewave signal at the output  66  of XOR gate  64  is then applied to the clock input of D-Type flip-flop  68 , which divides the signal into a 3 MHz squarewave. The 3 MHz output  70  from D-Type flip-flop  68  is applied to the clock input of D-Type flip-flop  71 , which further divides the signal into a 1.5 MHz squarewave signal. O-Type flip-flop  71  has two complementary outputs  72 ,  74 , where the first output  72  generates a first 1.5 MHz squarewave, which is applied to the clock input of D-Type flip-flop  73 . The second output  74  generates a second 1.5 MHz squarewave, which is 180 degrees out of phase with the a first 1.5 MHz squarewave. This signal is applied to the clock input of D-Type flip-flop  75  D-Type flip-flop  73  divides the first 1.5 MHz squarewave to a first 750 KHz squarewave signal, which is present at its output  84 . Similarly, D-Type flip-flop  75  divides the second 1.5 MHz squarewave to a second 750 KHz squarewave signal, which is present at its output  76  The first and second 750 KHz squarewaves are 90 degrees out of phase as a result of being clocked by the anti-phase first and second 1.5 MHz squarewaves.  
         [0034]    The series connected coils  22 ,  24  and capacitor  77  provide a parallel resonant circuit tuned to 750 KHz when the conductive core  26  is in its center position (i.e. voice coil is in the optimum operating region). The second 750 KHz squarewave at output  76  is filtered by capacitor  78  and resistor  80 , such that at point B at the terminal of resistor  80 , the second 750 KHz squarewave is converted to a 750 KHz sinusoidal signal of the same phase. Provided that the triangular conductive core  26  is in its center position, the phase of the 750 KHz sinusoidal signal does not change. The 750 KHz sinusoidal signal is then re-converted back to a 750 KHz squarewave by comparator circuit  82 , whereby if the phase has not been affected by the resonant circuit (i.e. core  26  is in its center position), the 750 KHz squarewave has the same phase as the signal output from D-Type flip-flop  75  Therefore, it will still have a 90-degree phase shift relative to the first 750 KHz signal generated by the output  84  of D-Type flip-flop  73 . It will be appreciated however, that the comparator circuit  82  has first and second complementary outputs  86 ,  88  that are 180 degrees out of phase. Hence, the first output  88  will have the same 90-degree phase shift relative to the first 750 KHz signal generated by the output  84  of D-Type flip-flop  73 , and the second output  86  will have a 270-degree phase shift relative to this signal (output from 84).  
         [0035]    EXOR logic gate  120  and low pass filter network  122  form a first phase comparator circuit, whilst EXOR logic gate  124  and low pass filter network  142  form a second phase comparator circuit. The first 750 KHz signal generated by the output  84  of D-Type flip-flop  73  is applied to the first input  130 ,  132  of both the first and second phase comparator network, respectively. Also, the first output  88  and the second output  86  from comparator  82  are applied to the second input  134 ,  136  of the first and second phase comparator network, respectively.  
         [0036]    Under these conditions, where the triangular core  26  is in the rest position, and the signals from the comparator  82  output  88  and the D-Type flip-flop  73  output  84  have a 90 degree phase difference, the first phase comparator XOR gate  120  output  138  will generate a squarewave signal with a 50% duty cycle. Therefore, the corresponding averaging applied to this signal by the low pass filter  122  will generate a DC voltage of 0 V at output  139 . Similarly, when the signals from the comparator  82  complementary output  86  and the output  84  from D-Type flip-flop  73  have a 270-degree phase difference, the second phase comparator XOR gate  124  output  140  will also generate a squarewave signal with a 50% duty cycle. Accordingly, this signal is averaged through the low pass filter  142 , wherein the averaged signal at output  144  is a DC voltage of approximately 0 V. Both DC outputs  139 ,  144  from the phase comparators are received by a differential amplifier  146 , which generates a difference signal based on the DC outputs  139  and  144 . This corresponding difference signal is the position control signal, and is amplified by amplifier  49 .  
         [0037]    Under the conditions where the speaker voice coil movement is centered about a position offset from its center position (i.e. optimum operating region centered about rest position), the change in inductance of the position sensor  20  varies with the resonance frequency of the parallel resonance circuit generated by the coils  22 ,  24  and capacitor  77 . This in turn causes an additional phase shift in the 750 KHz sinusoidal signal, at point B, relative to the first 750 KHz squarewave signal, which is present at the output  84  of D-Type flip-flop  73  The relative phase difference between these two signals will depart from 90-degrees (depending on direction of core  26  movement), which causes one output (e.g.  138 ) from one XOR gate (e.g.  120 ) to generate a squarewave signal with a duty cycle greater than 50%, whilst the other output (e.g.  140 ) from the other XOR gate (e.g.  124 ) generates a squarewave signal with a duty cycle less than 50%. DC averaging of the squarewave with a duty cycle greater than 50% will generate a positive DC voltage in proportion to the width of the pulses. Also, DC averaging of the squarewave with a duty cycle less than 50% will generate a lesser magnitude DC voltage in proportion to the width of the pulses. The DC voltages from the low pass filter  122 ,  142  outputs  139 ,  144  are received by the differential amplifier  146 , and a corresponding position control signal  48  is generated. The more the core  26  is displaced relative to its center position, the more the duty cycle of the squarewave signals is effected. Therefore, the magnitude difference between the DC voltages generated by averaging these squarewaves is increased. Hence, the position control signal  48  generated by the differential amplifier  146  increases. The generated position control signal  48  is directly proportional to the voice coil  34  and hence the core  26  displacement (see FIG. 1). This signal  48  is amplified, as indicated at  149 , and may then be applied to provide feedback to compensate for distortion as described, for example, in a co-pending application by the same applicant and also claiming priority from U.S. application No. 60/329,350.  
         [0038]    [0038]FIG. 4 illustrates an example of the mechanical construction of a speaker device  40  and the relative position of the acceleration sensor  42  and position sensor  20  As illustrated in the FIG. 4, the acceleration sensor  42  and position sensor&#39;s triangular conductive core  26  are connected to the bottom region of the voice coil bobbin  32 . The first and second inductance coils  22  (only one coil shown) are connected to a fixed (stationary) position or physical location on the speaker on either side of the triangular conductive core  26 . Consequently, as the voice coil  34  moves, the triangular conductive core  26  moves within the inductance coils  22 . Therefore, the position sensor  20  generates the electrical feedback control signal (or position control signal) necessary for distortion reduction. As shown in FIG. 4, the triangular conductive core  26  is connected to the bobbin  32  by means of bracket  44 . The acceleration sensor  42  also generates the electrical feedback control signal, which is linearly proportional to the movement of the voice coil  34  and bobbin  32 .  
         [0039]    Shaping the Position Sensor to Provide a Linear Output Voltage  
         [0040]    In accordance with a preferred aspect of the invention, a suitable conductive core  26  can be designed using empirical data obtained regarding the interaction of the material from which the conductive core is made with the other components of the loudspeaker. To begin, a regular, triangular-shaped conductive core is made from a selected conductive material such as a printed circuit board. The height of this triangle must be sufficient to extend over the entire maximum desirable stroke of the cone. After inserting the triangular element halfway between coils  22 ,  24 , the capacitor of FIG. 3 is adjusted to get zero volts of the circuit output  92 . The coils  22 ,  24  and triangular core  26  are installed in a designated speaker as the proximity of the speaker construction elements will help to determine what shape provides the desired output. A series of measurements must then be made covering the entire range of displacement.  
         [0041]    Referring to FIG. 5, there is illustrated in a graph, the outcome of a test using a regular triangular conductive core  26 . Specifically, in FIG. 5, output voltage in volts is plotted against displacement in inches. Despite the linearity of the width of the triangular conductive core  26  relative to distance from its base, the output voltage clearly departs from linear  
         [0042]    Only a portion of the triangular conductive core  26  influences the resonance frequency of the coils  22 ,  24  and the capacitor  77 . This portion is located between the two coils  22 ,  24 . Thus, there is a relationship between the width of the triangular conductive core  26  of the geometrical center of the coils  22 ,  24 , and system resonance.  
         [0043]    As the conductive core  26  being tested is a regular triangular shape, there is a linear relation between the width of that portion of the triangular conductive core  26  that is between the coils  22 ,  24  and the displacement of the triangular conductive core  26  from a reference position.  
         [0044]    Referring to FIG. 6, this relation is illustrated in a graph plotting the average width of that portion of the triangular conductive core that is between the coils  22 ,  24  against displacement of the triangular conductive core  26  from a rest position. No measurements are required to provide this graph, as the dimensions of the triangular conductive core  26  are known. As the conductive core  26  is of a regular triangular shape, the relationship between displacement and width is, of course, linear.  
         [0045]    Using the graphs of FIGS. 5 and 6, another graph, FIG. 7, may be plotted. The graph of FIG. 7 is generated by replacing the displacement axis of the graph of FIG. 6 with the corresponding output voltage determined by the graph of FIG. 5. For example, FIG. 5 indicates that a displacement of approximately −0.2 inches corresponds to an output voltage of approximately −2 volts. Referring to FIG. 6, a displacement of approximately −0.2 inches corresponds to a width of 0.6 inches. Thus, in FIG. 7, an output voltage of −2 volts corresponds approximately to a width of 0.6 inches.  
         [0046]    The position sensor  20  has a position sensor sensitivity S, which can be expressed in volts per inch. In the present example, the position sensor sensitivity is 6.8 volts per inch. Using this position sensor sensitivity, another graph similar to FIG. 7 can be plotted; however, in this graph the horizontal axis is not in volts but in inches. That is, by dividing the output voltage shown on the X axis of the graph of FIG. 7 by the position sensor sensitivity, the displacements corresponding to these output voltages can be determined  
         [0047]    Referring to FIG. 8, the width of a triangular conductive core in inches is plotted against these displacements. The graph of FIG. 8 has the same units along its X and Y axes as the graph of FIG. 6. However, the graph of FIG. 6 represents a triangle. Clearly, the graph of FIG. 8 represents a shape that is roughly triangular, but departs from the triangular as the width does not vary absolutely linearly with the displacement. Based on the graph of FIG. 8, a position sensor  20  can be designed in which the width varies according to the displacement in the manner shown in FIG. 8.  
         [0048]    Referring to FIG. 9, the output voltage generated by a position sensor  20  manufactured according to the specifications of the graph of FIG. 8 is plotted against the displacement of this position sensor  20 . As can be seen, the output voltage of this position sensor  26  varies substantially linearly with displacement.  
         [0049]    It is important to note that the foregoing method can be applied to design position sensors providing any one of a number of desired voltage outputs, and is not limited to merely providing linear outputs. Such non-linear outputs may be used to compensate for various sources of speaker non-linearity. One such source is the motor that drives the voice coil  34 . In the motor, a current i, flowing through the voice coil  34  generates a force F according to the following equation:  
         
       F=Bl·i  
     
         [0050]    where Bl is the force factor.  
         [0051]    However, Bl is not constant, but is a function of voice coil displacement X:  
           F=Bl ( x )· i    
         [0052]    As the displacement of the motor increases, the force Bl is significantly reduced to below what it should be, creating harmonic distortions. A typical relationship between Bl and displacement is illustrated in the graph of FIG. 10, which plots displacement against the ratio of actual force factor to the force factor when the voice coil  34  is at rest.  
         [0053]    The curve of FIG. 10 is parabolic. This is often, but not always, a good model of reality. Designers will sometimes want to know how the force factor really varies with the displacement. A position sensor designed in accordance with the present invention can help to provide this information  
         [0054]    [0054]FIG. 7 plots the relationship between the width of a triangular core and the output voltage. Specifically, using the relationship plotted in FIG. 7, a designer can decide on what output voltage is desired at each displacement position of the position sensor, and then can shape the conductive element such that the width at that position displacement is the width corresponding to the desired output voltage on the line plotted in FIG. 7. A designer may construct almost any conductive element, having almost any variation of width as a function of its displacement to obtain almost any transfer function (of course, the designer will be limited by the distance between the coils  22 ,  24  as the maximum width of the conductive element cannot exceed this distance). The procedure is much the same as in the case of a linear sensor. The only difference in the present example, it that the target transfer function is parabolic.  
         [0055]    Referring to FIG. 11, the rough shape of a conductive element required to obtain a parabolic transfer function is illustrated in a graph plotting width against displacement. The transfer function provided by this shape is shown on the graph of FIG. 12, which plots output voltage in volts against displacement. Alternatively, a parabolic transfer function can be obtained using a conductive element having the shape illustrated in the graph of FIG. 13, which plots displacement against width. The transfer function provided by the conductive element shape of FIG. 13 is illustrated in the graph of FIG. 14, which plots output voltage against displacement. The transfer function of FIG. 14 is inverted relative to the transfer function of FIG. 12. Thus, depending on the application, one of these transfer functions will require a voltage inverter and an associated circuit. Further, both of these transfer functions must be shifted to provide a transfer function similar to that shown in FIG. 10.  
         [0056]    Referring to FIG. 15 there is illustrated in a schematic diagram, a loudspeaker  102  having a motional feedback system  100  for reducing non-linear distortion introduced by the motor driving the voice coil. The loudspeaker  102  comprises a position sensor  108 . This position sensor has the configuration of the position sensor represented by the graph of FIG. 11. Accordingly, this position sensor  108  has an output voltage  
         (     V   ps     )     =     k   ·       Bl        (   x   )         Bl        (   0   )         ·     V   ps                             
 
         [0057]    [0057] 110  is transmitted to feedback network  112 , which also receives input audio signal  104 . Divider  112  then provides an output voltage  114 , which is amplified and converted to an audio current drive signal  106  by power amplifier  116  Audio current drive signal (I α ) is determined as follows  
       (       I   a     =     Input     V   ps         )                         
 
         [0058]    Thus, the force generated by the speaker motor structure is  
       F   =       Bl        (   x   )       ·     Input     V   ps                               
 
         [0059]    Recall, however, that  
         (     V   ps     )     =     k   ·       Bl        (   x   )         Bl        (   0   )                                 
 
         [0060]    By combining the two foregoing equations, one gets  
       F   =       Bl        (   0   )       ·     Input   k                             
 
         [0061]    Thus, the force generated by the speaker motor structure is a function of the input signal only, and the distortions are compensated for this solution is superior to the prior art solutions in that the prior art solutions require a special circuit inserted between the position sensor  108  and the divider  112  This additional circuit models the Bl(x) function. In contrast, or according to the present invention, the sensing and modeling are done by the same sensor, and modeling of Bl(x) is done with high precision for no extra effort or cost.  
         [0062]    Other variations and modifications of the invention are possible. For example, while the foregoing description has focused on position sensors that provide a linear or parabolic output relative to displacement, as described above, the potential output that can be provided by a position sensor according to the present invention is not limited to these two embodiments, that may be used to provide a wide range of different output voltages. Further, while the position sensor has been described in the context of loudspeakers, it will be appreciated by those skilled in the art that the position sensor may also be applied in other context.  
         [0063]    Also, while the present invention as described above is implemented using conductive cores, it will be appreciated by those skilled in the art that it may also be implemented using a ferromagnetic core. In general it is only required that the core affect the inductance in some way, by either increasing or decreasing it, so that the change in inductance can be determined, which in turn enables the degree of movement or deflection to be determined. If a ferromagnetic core is used, then increasing the width of the core will tend to increase inductance instead of diminishing it, requiring design modification. Further, while the above-described inductance-affecting capacity of the core is varied by varying the width, it will be appreciated by those skilled in the art that inductance-varying capacity may also be varied in other ways, such as, for example, by varying the composition or thickness of the core along its length, or by adding grooves to vary its resistance. All such modifications are within the sphere and scope of the invention as defined by the claims appended hereto.