Abstract:
A method for calibrating a magnetic field generator, including fixing one or more magnetic field sensors to a probe in known positions and orientations and selecting one or more known locations in the vicinity of the magnetic field generator. The magnetic field generator is driven so as to generate a magnetic field. The probe is moved in a predetermined, known orientation to each of the one or more locations, and signals are received from the one or more sensors at each of the one or more locations. The signals are processed to measure the amplitude and direction of the magnetic field, at the respective positions of the one or more sensors and to determine calibration factors relating to the amplitude and direction of the magnetic field in the vicinity of the magnetic field generator.

Description:
This application claims the benefit of U.S. Provisional Patent Application No. 60/016,908, filed May 6, 1996, which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to apparatus for generating and detecting electromagnetic fields, and specifically to non-contact, electromagnetic methods and devices for tracking the position and orientation of an object. 
     BACKGROUND OF THE INVENTION 
     Non-contact electromagnetic tracking systems are well known in the art, with a wide range of applications. 
     For example, U.S. Pat. No. 4,054,881, incorporated herein by reference, describes a tracking system using three coils to generate electromagnetic fields in the vicinity of the object. The fields generated by these three coils are distinguished from one another by open loop multiplexing of time, frequency or phase. The signal currents flowing in three orthogonal sensor coils are used to determine the object&#39;s position, based on an iterative method of computation. 
     Other electromagnetic tracking systems are described in U.S. Pat. Nos. 3,644,825, 3,868,565, 4,017,858 and 4,849,692, whose disclosures are likewise incorporated herein by reference. 
     U.S. Pat. No. 5,391,199, to Ben-Haim, which is incorporated herein by reference, describes a system for generating three-dimensional location information regarding a medical probe or catheter. A sensor coil is placed in the catheter and generates signals in response to externally applied magnetic fields. The magnetic fields are generated by three radiator coils, fixed to an external reference frame in known, mutually spaced locations. The amplitudes of the signals generated in response to each of the radiator coil fields are detected and used to compute the location of the sensor coil. Each radiator coil is preferably driven by driver circuitry to generate a field at a known frequency, distinct from that of other radiator coils, so that the signals generated by the sensor coil may be separated by frequency into components corresponding to the different radiator coils. 
     PCT patent publication No. WO96/05768, whose disclosure is incorporated herein by reference, describes a system that generates six-dimensional position and orientation information regarding the tip of a catheter. This system uses a plurality of non-concentric sensor coils adjacent to a locatable site in the catheter, for example near its distal end, and a plurality of radiator coils fixed in an external reference frame. The sensor coils generate signals in response to magnetic fields generated by the radiator coils, which signals allow for the computation of six location and orientation coordinates. 
     Radiator coils with cores are known in position sensing systems. The cores increase the field output of the coils, but they tend to distort the fields, and therefore reduce the accuracy of position detection. The theory of magnetic fields generated by radiator coils with cores is known in the art, as described, for example, by John David Jackson in  Classical Electrodynamics,  Second Edition (1975), pages 168-208, which is incorporated herein by reference. In practice, however, it is difficult to derive a theoretical model that will accurately predict the magnetic field generated by a coil with a core. 
     Ferrite cores are advantageous, because they have both high magnetic permeability (μ) and high resistivity (ρ). Due to the high resistivity, the cores can be used with a time-varying (AC) magnetic field without inducing eddy currents in the cores, which further distort and complicate the magnetic field. The Polhemus position-sensing system, as described, for example, in U.S. Pat. No. 4,017,858, uses such ferrite cores in its (AC) radiators. Ferrite materials are relatively expensive and fragile, however, making them impractical and uneconomic for use in sizes over about 5 cm in diameter. 
     Soft iron cores are also effective in increasing magnetic field output of a coil, but they cause serious distortion of AC magnetic fields due to eddy currents generated in the core by the coil. The Ascension position-sensing system, described in U.S. Pat. No. 4,849,692, is based on a DC magnetic field, and can therefore use soft iron cores in its DC radiator coils, since no eddy currents are generated by the DC field. 
     SUMMARY OF THE INVENTION 
     The accuracy and efficacy of electromagnetic tracking systems, such as those cited above, is generally dependent on precise knowledge of the distribution of the magnetic fields generated by the radiator coils. Although these fields may be calculated theoretically, based on the geometry of the coils, the actual magnetic fields typically differ from the theoretical models. For example, the fields may differ from the models due to small deviations in the manufacture of the coils. In the case of coils having a ferromagnetic core, the geometry and electrical and magnetic properties of the core must also be taken into account. There will typically be greater deviations from the theoretical models due, for example, to nonlinearities, hysteresis and eddy currents in the core, and to imprecise location of the core relative to the coils. These deviations may lead to inaccuracies in determining the position and orientation of the object being tracked. It would, therefore, be desirable to calibrate the radiator coils by precise measurement of the direction and amplitude of the magnetic field in the vicinity of the object to be tracked. 
     It is thus an object of some aspects of the present invention to provide a method and apparatus for calibrating electromagnetic radiator coils or other types of magnetic field generators. 
     In some aspects of the present invention, the field equations of an electromagnetic radiator coil are used to derive a parametric, theoretical model of the field, which is compared with calibration measurements of the field to determine accurate values of the parameters. 
     In one aspect of the present invention, the theoretical model takes into account perturbations of the field due to the effect of a ferromagnetic core in the radiator coil. 
     In another aspect of the present invention, the radiator coils are used as part of an object tracking system, such as a system for use in determining the position and orientation of a probe inside the body of a subject during a medical or surgical procedure. 
     In preferred embodiments of the present invention, apparatus for calibrating magnetic field generators comprises at least one sensor coil, fixed to a positioning device in a known geometrical relation. The positioning device, which may be of any suitable type known in the art, is adapted to position the at least one sensor coil in one or more known positions in a vicinity of the field generator being calibrated. The at least one sensor coil generates electrical signals in the presence of a time-varying magnetic field, which signals are analyzed to determine the direction and amplitude of the magnetic field at the positions of the coils. 
     In some preferred embodiments of the present invention, the at least one sensor coil comprises a plurality of sensor coils, preferably including three non-concentric coils, which are mutually substantially orthogonal, and are fixed in a predetermined mutual spacing. Non-concentric coils are advantageous in that they may more readily be wound in a small volume, preferably 1 mm 3  or less, desired for use in accordance with the present invention. 
     In some of these preferred embodiments, the coils are fixed in a substantially linear arrangement. Preferably the positioning device positions the coils successively in a plurality of positions along an axis defined by the arrangement of the coils. In one such preferred embodiment, the three non-concentric coils are fixed in a probe substantially as described in PCT patent application No. PCT/US95/01103, whose disclosure is incorporated herein by reference. 
     In other preferred embodiments of the present invention, the coils are fixed to respective faces of a cube. In one such preferred embodiment, six coils are respectively fixed to the six faces of the cube, such that the axis of each of the coils is orthogonal to the respective face to which it is fixed. Preferably, the positioning device positions the cube in a plurality of positions on a grid defined by the arrangement of the coils on the cube. 
     In preferred embodiments of the present invention, a method for calibrating a magnetic field generator comprises placing at least one sensor coil in one or more known positions and orientations in a vicinity of the field generator, driving the field generator to generate a time-varying magnetic field, and measuring the electrical signals generated by the at least one sensor coil, so as to determine the direction and amplitude of the magnetic field at the one or more known positions. The coil may have an air core or, preferably, a ferromagnetic core. 
     In some preferred embodiments of the present invention, wherein the field generator is substantially rotationally symmetrical about an axis thereof, the method for calibrating the field generator includes defining a calibration plane having a first axis defined by an axis of rotational symmetry of the field generator and a second axis chosen to be orthogonal to the first axis. Preferably the second axis is in a plane defined by the field generator. The at least one sensor coil is then placed in one or more known positions that are substantially within a quadrant of this plane, defined by the first and second axes, and the directions and amplitudes of the magnetic fields are determined in this quadrant. Due to the substantial symmetry of the field generator, the directions and amplitudes of the magnetic field determined in this quadrant are sufficient to determine the directions and amplitudes of the magnetic field in any other quadrant defined by choosing another second axis orthogonal to the first axis. 
     In a preferred embodiment of the present invention, the method for calibrating a field generator includes fixing three sensor coils to a positioning device in known, mutually substantially orthogonal orientations and in known positions in a non-concentric, substantially linear arrangement. The positioning device is used to place the coils successively in a plurality of known positions along a first axis defined by the arrangement of the coils. The electrical signal generated by each of the three sensor coils at each of the plurality of positions along this first axis is used to determine the amplitude of the component of the magnetic field projected along the direction of orientation of the respective sensor coil. Three such component amplitudes are thus determined at each of the plurality of positions, so that the magnetic field is completely determined along the first axis. The positioning device is then shifted to one or more additional axes, parallel to and in known displacement relative to the first axis, and the steps described above are repeated so as to determine the magnetic fields along these additional axes. 
     Alternatively, in another preferred embodiment of the present invention, fixing the three sensor coils comprises fixing a position sensing device including three sensor coils, substantially as described in the above-mentioned PCT patent application No. PCT/US95/01103. Position signals received from the device at each of the plurality of known positions in the vicinity of the field generator are compared with the actual, known position coordinates, so as to generate a calibration function. 
     In other preferred embodiments of the present invention, the at least one sensor coil is used to make additional measurements in both the calibration plane, as described above, and one or more additional planes, preferably having the same first axis as the calibration plane, but having different, respective second axes. Such additional measurements are useful in calibrating the field generator when the field may deviate from rotational symmetry, due, for example, to asymmetry and/or eccentricity of a ferromagnetic core within the radiator. 
     In still other preferred embodiments of the present invention, the at least one sensor coil is used to make measurements of the direction and amplitude of the magnetic field at a grid of points in the vicinity of the field generator. 
     There is therefore provided, in accordance with a preferred embodiment of the present invention, a method for calibrating a magnetic field generator, including: 
     fixing one or more magnetic field sensors to a probe in known positions and orientations; 
     selecting one or more known locations in a vicinity of the magnetic field generator; 
     driving the magnetic field generator so as to generate a magnetic field; 
     moving the probe in a predetermined, known orientation to each of the one or more locations; 
     receiving signals from the one or more sensors at each of the one or more locations; 
     processing the signals to measure the amplitude and direction of the magnetic field, at the respective positions of the one or more sensors; and 
     determining calibration factors relating to the amplitude and direction of the magnetic field in the vicinity of the magnetic field generator. 
     Preferably, fixing one or more magnetic sensors to a probe includes fixing sensor coils to the probe. Two or more sensor coils are preferably fixed to the probe, in orientations such that respective axes of the coils are mutually substantially orthogonal. 
     Preferably, fixing one or more magnetic sensors to a probe includes fixing three sensors to the probe, such that the positions of the sensors on the probe are substantially collinear. 
     Alternatively, fixing one or more magnetic sensors to the probe includes fixing sensors to a cube. 
     Preferably, selecting one or more known locations includes selecting a plurality of locations, and moving the probe includes moving the probe along an axis defined by the positions of the sensors on the probe and passing through two or more of the plurality of locations, preferably in steps of substantially equal length, such that the distance between any two of the sensors is substantially integrally divisible by the length of the steps. 
     Preferably, for calibrating a magnetic field generator that is substantially rotationally symmetrical, selecting the one or more known locations includes selecting one or more locations in a quadrant defined by the axis of rotational symmetry of the magnetic field generator and by a second axis in a plane defined by the magnetic field generator and normal to the axis of rotational symmetry, and moving the probe includes orienting the probe so that the one or more sensors are positioned in the plane. 
     Determining calibration factors preferably includes calculating theoretical values of the amplitude and direction of the magnetic field generated by the magnetic field generator at the one or more known locations; comparing the theoretical values to the amplitude and direction of the magnetic field measured at said locations; and computing arithmetic factors corresponding to the difference between the theoretical values and the measured amplitude and direction of the magnetic field at each such location. 
     Preferably, computing arithmetic factors includes fitting the theoretical values to the measured amplitude and direction of the field. 
     Preferably, calculating theoretical values includes deriving a theoretical model of the magnetic field in the presence of an air core within the magnetic field generator, and modifying the model to account for the presence of a ferromagnetic core within the magnetic field generator. 
     Alternatively or additionally, modifying the model includes determining a perturbation of the field due to a the core, preferably by determining a perturbation due to a nonlinearity of the core or, further additionally or alternatively, by determining a perturbation due to eddy currents in the core. 
     In a preferred embodiment, the method described above further includes fixing a magnetic-field-responsive position-sensing device to an object; placing the object in the vicinity of the magnetic field generator; receiving signals from the position-sensing device; processing the signals so as to calculate the position or orientation of the object; and applying the calibration factors so as to improve the accuracy of calculation of the position or orientation. 
     Preferably, calibrating the magnetic field includes storing the calibration factors in a memory associated with the radiator coil. 
     There is further provided, in accordance with a preferred embodiment of the present invention, apparatus for calibrating a magnetic field generator including: 
     a plurality of magnetic field sensors, which generate electrical signals in response to magnetic fields applied thereto by the field generator; and 
     a positioning device, for moving the sensors, 
     wherein the sensors are fixed to the positioning device in a substantially linear arrangement, and 
     wherein the positioning device has an axis of motion that is parallel to an axis defined by the substantially linear arrangement of the coils. 
     Preferably, the magnetic field sensors include sensor coils, which are fixed so that respective axes of the coils are mutually substantially orthogonal. 
     Preferably, the magnetic field sensors are fixed to the positioning device in a substantially linear arrangement. 
     Alternatively, the magnetic field sensors include sensors which are fixed to the faces of a cube, which is fixed to the positioning device. 
     Preferably, the sensor coils generate signals which are received by a computer which compares the signals to a theoretical model, so as to calibrate the magnetic field generator. 
     There is also provided, in accordance with a preferred embodiment of the present invention, a calibrated magnetic field generator, including: 
     at least one coil, which is driven to generate the magnetic field; and 
     an electronic memory circuit, associated with the at least one coil, for storing calibration factors relating to the field generated by the coil. 
     Preferably, the field generator includes a core inside the at least one coil, most preferably a ferromagnetic core. Preferably, the ferromagnet includes a ferrite, or alternatively, soft iron. 
     Preferably, the field generator includes an electronic memory circuit, most preferably an EPROM. 
     Preferably, the calibration factors relate the field generated by the at least one coil to a theoretical model thereof. 
     Preferably, the calibration factors include look-up tables. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will be more fully understood from the following detailed description of the preferred embodiments thereof, taken together with the drawings in which: 
     FIG. 1 is a schematic illustration of apparatus for calibrating a magnetic field generator, in accordance with a preferred embodiment of the present invention; 
     FIG. 2A is a schematic, sectional view of a magnetic field generator coil with an air core, for purposes of illustrating calibration of the coil, in accordance with a preferred embodiment of the present invention; 
     FIG. 2B is a schematic, sectional view of a magnetic field generator coil with a ferromagnetic core; for purposes of illustrating calibration of the coil, in accordance with a preferred embodiment of the present invention; 
     FIG. 3 is a schematic isometric illustration of a magnetic field generator coil with a ferromagnetic core, showing a coordinate system used in deriving a parametric model of the magnetic field due to the coil, in accordance with a preferred embodiment of the present invention; and 
     FIG. 4 is a graph illustrating a theoretical model of a magnetic field generated by the coil of FIG. 3; 
     FIG. 5 is a schematic isometric illustration of apparatus for calibrating a magnetic field generator, in accordance with another preferred embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Reference is now made to FIG. 1, which shows three sensor coils  20 ,  22 ,  24  for calibrating a magnetic field generator (not shown in FIG.  1 ). Coils  20 ,  22  and  24  are preferably of small size, each coil having a volume of approximately 1 mm 3 . The coils are fixed in a substantially linear arrangement to probe  26 , which is in turn fixed to positioning device  30 . Probe  26  and associated parts are preferably made from rigid plastic or other non-conducting substance, so as not to distort the lines of magnetic field. Coils  20 ,  22  and  24  are preferably oriented in predetermined, known orientations, which are mutually substantially orthogonal. In the presence of a time-varying magnetic field, electrical currents are induced in the coils, which are substantially proportional to the amplitudes of the components of the magnetic field along the coils&#39; respective axes at their respective positions. These signals are conveyed by wires  32  to signal processing apparatus  34 , which processes the signals to determine the direction and amplitude of the magnetic field. 
     In the preferred embodiment of the present invention shown in FIG. 1, positioning device  30  is an X-Y translation stage, which may be of any suitable type known in the art. In other preferred embodiments of the present invention, positioning device  30  may be an X-Y-Z translation device, or it may also include one or more rotation elements. Device  30  may be manually operated, motorized, or actuated using other means and methods known in the art, for example, by a robot. 
     The positions of coils  20 ,  22  and  24  on probe  26  define an axis of motion  36 , which is parallel to the Y-direction as illustrated in FIG.  1 . In a preferred embodiment of the present invention, positioning device  30  is adapted so as to move probe  26  along axis  36 . Preferably device  30  moves probe  26  in steps of constant size, such that the distance between any pair of coils  20 ,  22  and  24  is an integral number of the steps. In this way, each of coils  20 ,  22  and  24  is positioned at each point along the axis, for example point  38 , in turn, so that three substantially orthogonal components of the magnetic field are determined at each such point. 
     After magnetic fields have been measured at all desired points along axis  36 , positioning device  30  shifts probe  26  by a predetermined, known distance in the X-direction, and then measurements are repeated by moving the probe along the Y-direction, as described above. 
     In another preferred embodiment of the present invention (not shown in the figures), the three non-concentric coils  20 ,  22  and  24  are fixed in a probe substantially as described in the above-mentioned PCT patent application No. PCT/US95/01103. This probe is fixed to a positioning device, and is used in place of probe  26  and coils  20 ,  22  and  24  in calibrating a radiator coil, as described above. 
     In still other preferred embodiments of the present invention, one, two, four or more coils, in any suitable geometrical configuration, may be used to calibrate a magnetic field generator. The coils may be concentric or non-concentric. 
     FIG. 2A is a sectional view of a radiator coil  40  having an air core  43  used in generating magnetic fields. Radiator coil  40  is rotationally symmetrical about symmetry axis  42 . A second axis  46  is chosen to be orthogonal to symmetry axis  42 , wherein second axis  46  is preferably located in a plane  48  defined by coil  40 . Axes  42  and  46  define a quadrant  44  of a plane normal to plane  48 . It will be appreciated by those skilled in the art that the magnetic fields generated by radiator coil  40  will also be rotationally symmetrical about axis  42 . Thus, the directions and amplitudes of a magnetic field generated by coil  40 , determined in relation to a quadrant  44 , are substantially independent of the choice of second axis  46 . 
     Therefore, in preferred embodiments of the present invention, radiator coil  40  is calibrated by measuring the direction and amplitude of the magnetic field at one or more points in quadrant  44  defined by axes  42  and  46 . The measured values of direction and amplitude at the one or more points in this quadrant are then compared with theoretically calculated values, and any substantial differences between measured and theoretical values are recorded and used to determine calibration correction factors. In a preferred embodiment of the present invention, the calibration correction factors are stored electronically in a memory  47 , preferably comprising an EPROM, or other programmable microcircuit, associated with the radiator coil. The correction factors determined in relation to quadrant  44  are subsequently applied to calibrate the magnetic field in all quadrants above plane  48  of the coil. 
     It will be appreciated that the method described above may equally be applied to determine calibration correction factors in relation to quadrants below plane  48  of coil  40 . Furthermore, if the coil is additionally symmetrical under reflection in plane  48 , the correction factors determined in relation to quadrant  44  will themselves be sufficient to determine calibration correction factors in relation to quadrants below the plane. 
     FIG. 2B is a sectional view of a radiator coil  60  used in generating magnetic fields. Coil  60  is substantially similar to coil  40 , as described above, except that coil  60  contains a ferromagnetic core  62 . Generally ferromagnetic core  62  will be formed from a non-conductive material such as a ferrite, or a conductive material such as soft iron. Ferromagnetic core  62  is rotationally symmetrical about symmetry axis  42 . It will thus be appreciated that the magnetic fields generated by radiator coil  60  and ferromagnetic core  62  will also be rotationally symmetrical about axis  42 . Thus, the directions and amplitudes of a magnetic field generated by coil  60  with core  62 , determined in relation to a quadrant  44 , are substantially independent of the choice of second axis  46  defined as above, so long as the symmetry is maintained. 
     The presence of ferromagnetic core  62  in coil  60  significantly enhances the amplitude of the magnetic field produced at a given position, compared to the field produced if no core is in place. The enhancement of the amplitude of the field enlarges the region, known as the mapping volume, in which sensor coils, for example, as described in the above-mentioned U.S. Pat. No. 5,391,199, give a sufficiently strong signal to enable accurate position measurements to be made. 
     Although the ferromagnetic core  62  increases the mapping volume relative to the current applied to the coil  60 , the simple situation described above regarding the form of the magnetic field and the calibration of the coil in the air core case becomes more complicated when a ferromagnetic core is present. The presence of the ferromagnetic core  62  may cause the field to deviate significantly from theoretical models due to core parameters such as permeability, resistivity, and hysteresis. If, for example, the ferromagnetic core  62  has a generally finite resistivity, as in the case of a soft iron core, time-dependent magnetic fields will introduce eddy currents in the core, which will significantly perturb the field. Furthermore, if the core is not precisely symmetrical or is not precisely centered in coil  60 , the magnetic field will further deviate from the theoretical model. 
     Thus, in a preferred embodiment of the present invention, parameters such as permeability, resistivity, hysteresis, position, shape and dimensions of the ferromagnetic core  62  are used in deriving a theoretical model against which the field of coil  60  is calibrated. The model preferably further includes parameters such as the number of turns, current flow, and cross-sectional area of the radiator coil  60 . It will be understood that the above parameters of the coil and core are enumerated here by way of example, and other relevant parameters may similarly be included. The model is used to generate theoretical values of a vector magnetic field B (comprising components B r , B θ , B φ ) produced by radiator coil  60  and ferromagnetic core  62 . 
     To calibrate coil  60 , magnetic fields are measured at a plurality of points, preferably about 300 points, preferably as described above, and the measured magnetic field values obtained are compared with the theoretical values. Using multi-variable fitting methods known in the art, the measured data are used to calculate corrected, effective values of parameters such as permeability and an effective number of turns of the coil, for example. These effective parameter values may then be used in the theoretical model to calculate the magnetic field accurately anywhere in the mapping volume. 
     FIG. 3 is an isometric view of coil  60 , as shown in FIG. 2B, illustrating a coordinate system used in deriving the theoretical values of the magnetic field. Radiator coil  60  is assumed to comprise n turns of wire having a radius a and having a current I flowing in the wire. Ferromagnetic core  62  is assumed to be a sphere having a radius b and having a permeability μ. As described in the above-mentioned text by Jackson, the theoretical field added to radiator coil  60  at point  64  by the presence of core  62  is generally given by:                B   r     =       ∑     l   =   0     ∞            -     B   l              r     -     (     l   +   2     )              (     l   +   1     )              P   l          (     cos                 θ     )                   (   1   )                 B   θ     =       ∑     l   =   0     ∞            B   l          r     -     (     l   +   2     )                P   l          (     cos                 θ     )                   (   2   )                                
     
       
         Bφ=0   (3)  
       
     
     where P l (cosθ) and P l ′(cosθ) are Legendre polynomials and their derivatives respectively, and 
     
       
         B l =0 when l is even  
       
     
     
       
         
           
             
               
                 
                   
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     where the magnetic dipole moment m of the coil is given by:              m   =       n                 π                   a   2        I     c             (   5   )                                
     where c is the velocity of light, and                  b   l     =           (     -   1     )     l            (       2      l     +   1     )     !!           2   l          l   !                
        and           (   6   )                 c   l     =             (     -   1     )     l            (       2      l     +   1     )     !!           2   l            (     l   +   1     )     !         ·       (       2      l     +   2     )         2      l     +   1                 (   7   )                                
     and (2l+1)!!≡(2l+1)(2l−1)(2l−3). . . ×5×3×1. 
     These equations are preferably modified, using mathematical methods known in the art, for example, perturbation theory, to account for such effects as variations in permeability, eddy currents, hysteresis and other deviations of coil  60  and core  62  from theoretical behavior, as described above. The modified equations may be in the form of analytical solutions, similar to equations (1) through (4) and equations (6) and (7) above, with suitable changes. Alternatively, they may take the form of a numerical solution, calculated by a computer, with results dependent on the variably values of the coil and core parameters. 
     FIG. 4 is a graph showing a cross section of the theoretical field added by the ferromagnetic core  62 , at a distance of 25 cm from the center of the core in coil  60 , as shown in FIG.  2 B. The field is calculated using equations (1) through (7), above, by inserting typical values for the parameters: μ=1000, a=5 cm and b=4.5 cm. Vertical axis  70  represents the fractional increase in the magnetic field magnitude, |B|, compared to the field with an air core, and horizontal axis  72  represents the angle θ measured in radians. It will be appreciated that changes in the values of the parameters used in the equations will result in changes in the shape of the curve in FIG.  4 . 
     To calibrate coil  60 , magnetic fields are measured at a plurality of points, using the system shown in FIG. 2A, for example. Using the fitting methods described above, parameters including an effective number of turns and an effective permeability are derived to give an optimal fitting of the curve shown in FIG. 4 to the measured values. These parameters are used in the equations above to calculate the magnetic field anywhere in the mapping volume. Alternatively, the parameters may be inserted into a numerical model for this purpose. 
     In preferred embodiments of the present invention, calibration of coil  40  with air core  43 , or coil  60  with ferromagnetic core  62 , is performed using the apparatus shown in FIG. 1, wherein coils  20 ,  22  and  24  on probe  26  are scanned mechanically by positioning device  30  through the one or more points in quadrant  44 , as described above. 
     In an alternative preferred embodiment of the present invention, not shown in the figures, a two-dimensional array of sensor coils, in predetermined, known positions and orientations, is used to calibrate a magnetic field generator. For calibrating a rotationally symmetrical field generator, such as coil  40  with air core  43  shown in FIG. 2A, or coil  60  with ferromagnetic core  62  shown in FIG. 2B, the array is preferably positioned so that all the sensor coils in the array are located in quadrant  44 . Thus, calibration correction factors may be determined substantially simultaneously for a substantial region of interest in the vicinity of the coil. 
     FIG. 5 shows yet another alternative preferred embodiment for use in calibrating a magnetic field generator (not shown in FIG.  5 ), in accordance with a preferred embodiment of the present invention, comprising a cube  92  in which six sensor coils  80 ,  82 ,  84 ,  86 ,  88 ,  90  are fixed. The coils preferably have a diameter of approximately 1 mm and a height of approximately several millimeters, but they are enlarged in FIG. 5 for clarity. The coils are preferably fixed to the sides of a cube  92  so that: the axes of coils  80 ,  86  are substantially collinear and lie generally parallel to the X-direction shown in FIG. 5; the axes of coils  82 ,  88  are substantially collinear and lie generally parallel to the Z-direction; and the axes of coils  84 ,  90  are substantially collinear and lie generally parallel to the Y-direction. The three aforementioned axes are substantially orthogonal, and the coils are fixed to the sides of the cube so that the three axes intersect generally at the center of the cube. 
     Cube  92  has an edge length of approximately 3 cm, and the center-center distance of collinear coils is generally 2 cm. Cube  92  and associated parts are preferably made from rigid plastic or other non-conducting material, so as not to distort the magnetic field. In the presence of time-varying magnetic fields, signals from the coils are conveyed by wires  96  to signal processing equipment  34  (not shown in FIG.  5 ). 
     The centers of the coils  84  and  90  on cube  92  define an axis  98 ; the centers of the coils  80  and  86  on cube  92  define an axis  104 ; and the centers of the coils  82  and  88  on cube  92  define an axis  102 . In a preferred embodiment of the present invention, cube  92  is set in positioning device  30  so that the edges of the cube are generally parallel to the X-, Y-, and Z-directions, and positioning device  30  moves cube  92  along axis  98 . Preferably device  30  moves cube  92  in steps of constant size, such as 3 cm. 
     After magnetic fields have been measured at all desired points on axis  98 , positioning device  30  shifts cube  92  by a predetermined, known distance such as 1 cm along axis  102 , and then measurements are repeated by moving the probe parallel to the Y-direction, as described above. After magnetic fields have been measured at all desired points in the plane defined by axes  98  and  102 , positioning device  30  shifts cube  92  by a predetermined, known distance such as 1 cm along axis  104 , and then measurements are repeated by moving the probe parallel to the Y-direction, as described above. In this way coils  80  or  86 , and coils  82  or  88 , and coils  84  or  90 , are positioned at each point, for example point  100 , in turn, so that three substantially orthogonal components of the magnetic field are determined at each such point. 
     In some preferred embodiments of the present invention, radiator coil  40  with air core  43 , or radiator coil  60  with ferromagnetic core  62 , is used in a system for tracking the position and/or orientation of an object (not shown in the figures) in a vicinity of the coil. Preferably position-sensing coils are placed on or adjacent to this object, and generate electrical signals in response to a magnetic field generated by coil  40  or  60 . The calibration correction factors determined in accordance with the above method are then applied to the electrical signals received from the position-sensing coils, so as to track the position and orientation of the object with greater accuracy. 
     In some such preferred embodiments of the present invention, the object being tracked is a catheter, for example, as described in the above-mentioned PCT patent application 01103 or in U.S. Pat. No. 5,391,199. Preferably, sensor coils  20 ,  22  and  24 , which are used to calibrate radiator coil  40  or  60 , are substantially similar to position-sensing coils adjacent to the distal end of the catheter. 
     In one such preferred embodiment of the present invention, signals received from these position-sensing coils are used to determine uncorrected position coordinates of the object, based on theoretical values of the amplitude and direction of the magnetic field generated by coil  40  or  60 . Calibration correction factors determined for the position indicated by these uncorrected coordinates are applied so as to calculate corrected values of the magnetic field amplitude and direction in the vicinity of the object. The corrected magnetic field amplitude and direction are then used to find corrected position coordinates of the object. 
     The calibration correction factors determined in accordance with preferred embodiments of the present invention are preferably stored in the form of a look-up table, comprising additive or multiplicative factors, which are applied in calculating corrected values of the magnetic field amplitude and direction and/or corrected position coordinates of the object. Correction factors for all points within a region in the vicinity of a field generator, wherein the direction and amplitude of the field have been measured at a plurality of such points, may be determined by methods of interpolation and curve fitting known in the art. 
     It will be appreciated that the preferred embodiments described above are cited by way of example, and the full scope of the invention is limited only by the claims.