Abstract:
An apparatus for measuring alternating current in a conductor comprises first and second coils a 1   x , d 1   x  having substantially the same turns-area product and substantially parallel axes and located on the circumference of a circle with the first coil having its axis tangential to the circle and the second coil having its axis radially of the circle, and third and fourth coils a 1   y , d 1   y  also having substantially the same turns-area product and substantially parallel axes, the third and fourth coil means being located on the circumference of the same circle close to the first and second coil means respectively but having their axes orthogonal thereto. The coils are mounted on a support means configured to allow a conductor to be introduced into the centre of the said circle with the axis of the conductor normal to the plane containing the coils. The first and second coils are connected in series in anti-phase and the third and fourth coil means are connected in series in anti-phase, and the alternating current in the conductor is derived as a function of the voltages induced in the series-connected first and second coils and the series-connected third and fourth coils. Further coils are provided for interference suppression and signal enhancement.

Description:
BACKGROUND OF THE INVENTION 
   This invention relates to an apparatus for measuring an alternating current flowing in an electric cable, for example an a.c. mains cable. 
   SUMMARY OF THE INVENTION 
   The present state of the art of measuring current in two or three core round cables is described in U.S. Pat. No. 5,652,506. Part of the coil arrangement used in this prior apparatus is shown in  FIG. 1 . 
   Eight identical wire-wound coils are used in total in the previous arrangement. For clarity sake, only four of these are shown in  FIG. 1 . The four coils shown pick up the x component Hx of magnetic field only from the cable  10 . The other four identical coils, not shown in this diagram, are located at the same centres as the four shown; however they are wound in a plane which is normal to the plane of the four shown in order to pick up the y component Hy of magnetic field only. 
   Coil  1  and coil  2  are the main pickup coils for the Hx component of magnetic field and are connected in phase addition. Coil  3  and coil  4  are connected in phase opposition to coil  1  and coil  2 . Coil  3  and coil  4  are used only to reduce the pickup of stray magnetic fields from other possible interfering current sources which are external to the coil arrangement. 
   The magnetic field pick-up of coil  1  and coil  2  from the current source to be measured is larger than the pickup by coil  3  and coil  4  since coils  3  and  4  are further away from the current source. Thus, when the output from coils  3  and  4  is subtracted from coils  1  and  2  the result is not zero. Therefore a voltage pickup proportional to the Hx component of magnetic field from the current source is present at the input to the amplifier AMP 1 . 
   For sources external and further away from the coil arrangement, however, the magnetic field created is much more uniform in magnitude and direction in the vicinity of the coils and the pickup from coils  3  and  4  almost completely cancels out the pickup from coils  1  and  2 , significantly reducing any errors incurred due to other interfering current sources in the vicinity of the apparatus. 
   At the output of the amplifier AMP 1  there is a voltage Vx which is proportional to the magnetic field component Hx created by the current flowing out in one conductor of the cable  10  and returning in the other. 
   The other four coils, picking up the Hy component, are connected identically to the four shown and are amplified separately by a similar amplifier AMP 2 , not shown. At the output of AMP 2 , therefore, a voltage Vy exists which is proportional to the Hy component of magnetic field. 
   It is useful at this stage to examine the Hx and Hy components of magnetic field created by a current I flowing out in one conductor of the cable  10  and returning in a second conductor of the cable, located a distance d away from the first conductor, as shown in  FIG. 2 . For the present it is assumed that the two conductors do not twist or rotate as they extend along the length of the cable. 
   Since the cable is round, no information is available as to the orientation angle θ of the coils to the conductors and θ is treated as a variable. Under these circumstances 
   
     
       
         
           
             
               
                 Hx 
                 = 
                 
                   
                     Id 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
           
             
               
                 Hy 
                 = 
                 
                   
                     Id 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             d 
                             2 
                           
                           
                             8 
                             ⁢ 
                             
                               r 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
   
   Referring back to  FIG. 1 , if coil  1  is at an angle θ to the current-carrying conductors, then coil  2 , which is located diametrically opposite to coil  1 , has an angle θ+180° to the conductors and it is seen from equation (1) that Hx is identical for angles θ and θ+180°. When coils  1  and  2  are connected in phase addition, therefore, the pickup voltage is double the pickup voltage of one of these coils on its own. 
   A similar analysis applies to the other coils picking up the Hy component of magnetic field as given by equation (2). 
   If the magnitude of the magnetic field |H| is then computed from the Hx and Hy components given by equations (1) and (2) respectively, the following result is obtained: 
   
     
       
         
           
             
               
                 
                    
                   H 
                    
                 
                 = 
                 
                   
                     
                       
                         Hx 
                         2 
                       
                       + 
                       
                         Hy 
                         2 
                       
                     
                   
                   = 
                   
                     
                       Id 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           r 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       1 
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
   
   Now Vx, the output voltage of AMP 1 , is proportional to Hx, and Vy, the output voltage of AMP 2 , is proportional to Hy. 
   V is evaluated from Vx and Vy, as follows: 
   
     
       
         
           
             
               
                 V 
                 = 
                 
                   
                     
                       
                         Vx 
                         2 
                       
                       + 
                       
                         Vy 
                         2 
                       
                     
                   
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                          
                         H 
                          
                       
                     
                     = 
                     
                       
                         KId 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       4 
                                       ⁢ 
                                       r 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
   
   Where K is a constant that depends on the area and number of the coil turns and the amplifier gain. 
   The computed voltage V is a maximum at θ=90° and is given by 
                   V   ⁢           ⁢   max     =     KId     2   ⁢           ⁢   π   ⁢           ⁢       r   2     ⁡     (     1   -       d   2       4   ⁢     r   2           )                   Eq   ⁢           ⁢     (   5   )                 
and V is a minimum when θ=0°
 
   
     
       
         
           
             
               
                 
                   V 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   min 
                 
                 = 
                 
                   KId 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               d 
                               2 
                             
                             
                               4 
                               ⁢ 
                               
                                 r 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
   
   The average value of V is approximately 
   
     
       
         
           
             
               
                 V 
                 = 
                 
                   KId 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       2 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
   
   Equation (7) is the equation used to evaluate the current I, since K and r are known, and d can be estimated fairly accurately in the previously patented technique. 
   For I fixed, however, there is an inherent variation in the measured voltage V as the coil positions around the cable vary as indicated by equations (5) and (6). The maximum variation from equation (7) depends on the value of d 2 /4r 2 . For d/r=½ there is a maximum variation (error) of ±6.3%. For d/r&lt;½ the variation (error) is smaller in theory. In practice, however, the best accuracy that can be achieved with this previous apparatus, in round cables, is of the order of ±11%. 
   The reason the performance is worse than indicated by equation (4) is primarily due to the fact that in round cables the conductors twist as they extend along the length of the cable. It was assumed in deriving equation (4) that the conductors stay parallel and straight along the cable. 
   Equation (4) indicates that the maximum pickup occurs when the angle θ=90° and the minimum pickup occurs when 0=0°. It is found that this effect reverses itself when the conductors are twisted beyond a certain limit, causing the maximum pickup to occur at θ=0° and the minimum to occur at θ=90°. The variation in pickup of this previous apparatus as the coils move round the twisted conductors depends on the rate of twist, the spacing d, and the distance to the coils r. 
   If V is computed from V=√{square root over (V x   2 +V y   2 ))} for the coil arrangement shown in  FIG. 1 , then variations in readings as the coils rotate round the cable vary from cable to cable but can exceed ±15%. 
   The coil arrangement shown in  FIG. 1  is therefore limited in its accuracy, and it is an object of the invention to provide a new apparatus which is capable of giving greater accuracy. 
   This object is met by the invention claimed in claim  1 . Preferred embodiments of the invention are claimed in the dependent claims. 
   In this specification the axis of a coil means that direction relative to the coil which, when orientated parallel to the direction of a fluctuating magnetic field passing through the coil, would provide the maximum induced voltage in the coil for that magnetic field. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     An embodiment of the invention will now be described, by way of example, with reference to the accompanying drawings, in which: 
       FIG. 1 , previously described, shows part of a prior art apparatus. 
       FIG. 2  is a diagram to assist in understanding the operation and limitations of the prior art. 
       FIG. 3  is a schematic diagram of the coil arrangement of a basic embodiment of the present invention. 
       FIG. 4  is a diagram of the embodiment of  FIG. 3  illustrating the effect of an interfering source. 
       FIG. 5  is a further development of the embodiment of  FIG. 3 . 
       FIG. 6  shows an embodiment of the invention with interference suppression. 
       FIG. 7  is a circuit diagram of the embodiment of  FIG. 6 . 
       FIG. 8  shows an alternative construction for the orthogonal coils of  FIG. 6 . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   As described, coils  1  and  2  of the apparatus shown in  FIG. 1  pick up the Hx component of magnetic field. These coils are also present in the apparatus shown in  FIG. 3 , which is a circuit diagram of the coil arrangement of a basic embodiment of the present invention, but they are referred to there as coils a x  and b x . The apparatus of  FIG. 3  also includes two further coils c x  and d x  which also pick up the Hx component of magnetic field. These two further coils c x  and d x  are located at the same distance as coils a x  and b x  from the centre of the cable  10  but are located above and below the cable so that a line joining coil c x  to coil d x  is at an angle of 90° to a line joining coil a x  and coil b x . Thus all four coils a x  to d x  lie in a plane which is perpendicular to the axis of the cable  10  on the circumference of a notional circle of radius r whose centre is coaxial with the cable  10 . All four coils a x  to d x  have substantially parallel axes; thus the axes of coils a x  and b x  are tangential to the notional circle while the axes of coils c x  and d x  extend radially of the notional circle. All four coils a x  to d x  have substantially the same turns-area product. 
   Thus, with reference to  FIG. 2 , if coil a x  is at an angle θ to the two conductors of the cable  10 , then coil d x  is at 90°+θ, coil b x  is at 180°+θ and coil c x  is at 270’+θ. The advantage of this apparatus over the previous one, if the coils are connected in series in the correct polarity in the manner to be described, is that the pickup repeats itself every 90° of rotation and so the pickup at θ=0° and θ=90° are identical. In the previous apparatus maximum variation between reading occurred at θ=0° and 90°. 
   The variation in pickup of the Hx component of magnetic field that occurs at each of the four coils a x  to d x  located round a parallel pair of conductors, as shown in  FIG. 3 , is now examined with the use of equation (1). 
   If coil a x  is at an angle θ to the conductor pair then Hx at coil a x  is given by equation (1) as: 
   
     
       
         
           
             
               
                 
                   Hx 
                   ⁡ 
                   
                     ( 
                     
                       a 
                       x 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
   
   For coil d x  its angle is 90+θ to the conductors and the Hx component there, again from equation (1), is 
   
     
       
         
           
             
               
                 
                   Hx 
                   ⁡ 
                   
                     ( 
                     
                       d 
                       x 
                     
                     ) 
                   
                 
                 = 
                 
                   - 
                   
                     
                       Id 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           r 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       4 
                                       ⁢ 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
   
   For coil b x  its angle is 180°+θ to the conductors and therefore 
   
     
       
         
           
             
               
                 
                   Hx 
                   ⁡ 
                   
                     ( 
                     
                       b 
                       x 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       
                                           
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
   
   For coil (c x ) its angle is 270°+θ to the conductors giving 
   
     
       
         
           
             
               
                 
                   Hx 
                   ⁡ 
                   
                     ( 
                     
                       c 
                       x 
                     
                     ) 
                   
                 
                 = 
                 
                   - 
                   
                     
                       Id 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           r 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       4 
                                       ⁢ 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 
                                   cos 
                                   2 
                                 
                                 
                                     
                                 
                               
                               ⁢ 
                               θ 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
   
   Coils a x  to d x  are all connected in series, as shown in  FIG. 3 , in the following polarities. Coils a x  and b x  have an in-phase connection and coils c x  and d x  are connected in anti-phase to coils a x  and b x  so that the voltage V(x) induced in the series connection of coils a x  to d x  is proportional to:
 
Hx(a x )+Hx(b x )−Hx(c x )−Hx(d x ).
 
   Substituting for the values of these magnetic fields from equations (8), (9), (10), and (11) and simplifying gives 
   
     
       
         
           
             
               
                 
                   
                     Hx 
                     ⁡ 
                     
                       ( 
                       
                         a 
                         x 
                       
                       ) 
                     
                   
                   + 
                   
                     Hx 
                     ⁡ 
                     
                       ( 
                       
                         b 
                         x 
                       
                       ) 
                     
                   
                   - 
                   
                     Hx 
                     ⁡ 
                     
                       ( 
                       
                         c 
                         x 
                       
                       ) 
                     
                   
                   - 
                   
                     Hx 
                     ⁡ 
                     
                       ( 
                       
                         d 
                         x 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       4 
                       ⁢ 
                       Id 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 d 
                                 4 
                               
                               
                                 16 
                                 ⁢ 
                                 
                                   r 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       
                                         r 
                                         2 
                                       
                                       
                                           
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                       [ 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             
                               d 
                               2 
                             
                             
                               r 
                               2 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
   
   The arrangement of  FIG. 3  also includes four further coils a y , b y , c y  and d y  each having substantially the same turns-area product as each of the coils a x  to d x . Each coil a y  to d y  is placed, as far as is physically practical, at the same location as a corresponding one of the coils a x  to d x , but its axis is rotated through 90° so that the axes of the coils a y  and b y  are tangential to the notional circle and the axes of the coils c y  and d y  extend radially of the notional circle. Thus four pairs a x /a y , b x /b y , c x /c y  and d x /d y  of closely positioned coils are present with the coils in each pair having substantially orthogonal axes. This close positioning of the pairs of coils a x /a y , etc. at the same location can be achieved, for example, using orthogonal pairs of coils as shown in FIG. 9 of U.S. Pat. No. 5,652,506 but an alternative construction using PCB technology will be described later. As will be evident to the reader, each of the coils a y  to d y  is orientated to pick up the y component Hy of the magnetic field generated by the cable  10  at the respective location. 
   Let Hy(a y ) be the y component of magnetic field picked up by the coil a y . 
   Let Hy(b y ) be the y component of magnetic field picked up by the coil b y . 
   Let Hy(c y ) be the y component of magnetic field picked up by the coil c y . 
   Let Hy(d y ) be the y component of magnetic field picked up by the coil d y . 
   From equation (2) 
   
     
       
         
           
             
               
                 
                   Hy 
                   ⁡ 
                   
                     ( 
                     
                       a 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             d 
                             2 
                           
                           
                             8 
                             ⁢ 
                             
                               r 
                               2 
                             
                           
                         
                       
                       ] 
                     
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             
                               d 
                               2 
                             
                             
                               r 
                               2 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
           
             
               
                 
                   Hy 
                   ⁡ 
                   
                     ( 
                     
                       d 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             cos 
                             2 
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             d 
                             2 
                           
                           
                             8 
                             ⁢ 
                             
                               r 
                               2 
                             
                           
                         
                       
                       ] 
                     
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             
                               d 
                               2 
                             
                             
                               r 
                               2 
                             
                           
                           ⁢ 
                           
                             cos 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
           
             
               
                 
                   Hy 
                   ⁡ 
                   
                     ( 
                     
                       b 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             d 
                             2 
                           
                           
                             8 
                             ⁢ 
                             
                               r 
                               2 
                             
                           
                         
                       
                       ] 
                     
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             
                               d 
                               2 
                             
                             
                               r 
                               2 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
           
             
               
                 
                   Hy 
                   ⁡ 
                   
                     ( 
                     
                       c 
                       y 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     Id 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         [ 
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             θ 
                           
                           - 
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               d 
                               2 
                             
                             
                               8 
                               ⁢ 
                               
                                 r 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     d 
                                     2 
                                   
                                   
                                     4 
                                     ⁢ 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               
                                 d 
                                 2 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
   
   These four coils a y to d   y  picking up the Hy component of magnetic field are connected in series with the same polarities as the four coils a x  to d y  picking up the Hx component; i.e. coils a y  and b y  are connected in phase and coils c y  and d y  are connected in antiphase to coils a y  and b y . To avoid over-complicating  FIG. 3  the connections between the coils a y  to d y  are not shown in that figure. 
   A voltage V(y) is therefore induced in the series connection proportional to
 
Hy(a y )+Hy(b y )−Hy(c y )−Hy(d y ).
 
   Substituting for the values of these magnetic fields from equation (13) to (16) and simplifying gives 
   
     
       
         
           
             
               
                 
                   
                     Hy 
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             a 
                             y 
                           
                           ) 
                         
                         + 
                         
                           Hy 
                           ⁡ 
                           
                             ( 
                             
                               b 
                               y 
                             
                             ) 
                           
                         
                       
                     
                   
                   - 
                   
                     Hy 
                     ⁡ 
                     
                       ( 
                       
                         c 
                         y 
                       
                       ) 
                     
                   
                   - 
                   
                     Hy 
                     ⁡ 
                     
                       ( 
                       
                         d 
                         y 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       Id 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                             - 
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 d 
                                 4 
                               
                               
                                 16 
                                 ⁢ 
                                 
                                   r 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                       
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       4 
                                       ⁢ 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       4 
                                       ⁢ 
                                       r 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                           ] 
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
   
   V is now evaluated as: 
                       V   =           V   ⁡     (   x   )       2     +       V   ⁡     (   y   )       2                     =     K   ⁢           [       Hx   ⁡     (     a   x     )       +     Hx   ⁡     (     b   x     )       -     Hx   ⁡     (     c   x     )       -     Hx   ⁡     (     d   x     )         ]     2     +       [       Hy   ⁡     (     a   y     )       +     Hy   ⁡     (     b   y     )       -     Hy   ⁡     (     c   y     )       -     Hy   ⁡     (     d   y     )         ]     2                         Eq   ⁢           ⁢     (   18   )                 
where K is a constant, as before.
 
   Substituting in equation (18) from equations (12) and (17), and simplifying, gives 
   
     
       
         
           
             
               
                 V 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       KdI 
                     
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     1 
                     
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   d 
                                   4 
                                 
                                 
                                   16 
                                   ⁢ 
                                   
                                     r 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             
                               d 
                               4 
                             
                             
                               r 
                               4 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           
                             θcos 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
   
   The minimum value of V occurs when θ=45° and this value is given by 
   
     
       
         
           
             
               
                 
                   V 
                   MIN 
                 
                 = 
                 
                   
                     2 
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               d 
                               4 
                             
                             
                               16 
                               ⁢ 
                               
                                 r 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
   
   The maximum value of V occurs when θ=0° or θ=90° and this maximum value is given by 
   
     
       
         
           
             
               
                 
                   V 
                   MAX 
                 
                 = 
                 
                   
                     2 
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         r 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               d 
                               4 
                             
                             
                               16 
                               ⁢ 
                               
                                 r 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
   
   The mean value is approximately 
   
     
       
         
           
             
               
                 
                   V 
                   AV 
                 
                 = 
                 
                   
                     2 
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       2 
                     
                   
                 
               
             
             
               
                 Eq 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
   
   Except for a factor of four, equation (22) of the new apparatus is exactly the same as equation (7) of the previous apparatus. However, the variation from maximum to minimum of the previous apparatus as the sensor is rotated depends on the magnitude of d 2 /4r 2 , whereas for the new coil arrangement it depends on the magnitude of d 4 /16r 4 . Thus, for d/r=½ the reading varies by ±6.3% for the previous apparatus whereas the new apparatus, with d/r=½, varies by only ±0.4%. 
   The improvement using the new coil arrangement as given by equation (19) compared to the previous apparatus as given by equation (4) is strictly true only for conductors which do not twist as they extend along the length of the cable. However, the new apparatus is far less prone to errors caused by cable rotation or conductor twisting with variations of &lt;2% recorded in V as the cable is rotated by 360° for this new coil arrangement. The previous apparatus records variations of 15% or larger where the same cable is rotated in the jaws of the instrument. 
   In the previous coil arrangement, as shown in  FIG. 1 , coil  3  and coil  4  are connected in anti-phase with coil  1  and coil  2 , and their purpose is to reduce the pickup of interference from the other current sources in the vicinity of the meter. The interference pickup of the coil arrangement shown in  FIG. 3  is now examined with reference to  FIG. 4 . 
     FIG. 4  shows the coils a x  to d x  located on the circle of radius r. Also shown is a cable  10  carrying a current I located at two possible positions, position A or position B. Position A is the location of the cable when a measurement of its current I is made. Position B shows the same cable, located a distance r I  from the centre of the circle, carrying the same current I but exterior to the coil arrangement, where it is acting as an interfering source. The interference suppression S of the coil arrangement is defined as: 
   
     
       
         
           
             
               
                 S 
                 = 
                 
                   
                     Pickup 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     in 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     position 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     B 
                   
                   
                     Pickup 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     in 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     position 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
   
   The smaller the value of S the better the suppression. The pickup in position A is given by equation (22) 
   
     
       
         
           
             Pickup 
             ⁢ 
             
                 
             
             ⁢ 
             in 
             ⁢ 
             
                 
             
             ⁢ 
             position 
             ⁢ 
             
                 
             
             ⁢ 
             A 
           
           = 
           
             
               2 
               ⁢ 
               KdI 
             
             
               π 
               ⁢ 
               
                   
               
               ⁢ 
               
                 r 
                 2 
               
             
           
         
       
     
   
   It may be shown that the pickup in position B is given by 
   
     
       
         
           
             
               
                 
                   Pickup 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   in 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   position 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
                 = 
                 
                   
                     6 
                     ⁢ 
                     
                       r 
                       2 
                     
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           I 
                           4 
                         
                         - 
                         
                           r 
                           4 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
           
             
               
                 
                   Therefore 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   S 
                 
                 = 
                 
                   
                     3 
                     ⁢ 
                     
                       r 
                       4 
                     
                   
                   
                     
                       r 
                       I 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             r 
                             4 
                           
                           
                             r 
                             I 
                             4 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
   
   It is seen from equation (25) that the smaller the value of 
           r     r   I           
the better the suppression.
 
   If it is assumed, due to coil and apparatus housings, that the closest an interference source can get to the coil arrangement is r I =2r, then the maximum value of S from equation (25) is S=0.2 or 20%. As the interfering source moves further away S decreases fairly rapidly, with S=4% for r I =3r and S=1% for r I =4r. This maximum interference value of 20% is in general unacceptable and is reduced significantly by employing the following technique. 
   It is noted from equation (24) that the interference pickup is primarily proportional to r 2  where r is the distance of the coils from the centre. Consider therefore the situation as shown in  FIG. 5 . In this arrangement, two sets of coils are used to pickup the Hx component of magnetic field, an inner set a 1   x  to d 1   x  located at 90° intervals around the circumference of a circle of radius r, and an outer set a 2   x  to d 2   x  located at 90° intervals around the circumference of a circle of radius r 2  coaxial with the first circle. The four inner coils a 1   x  to d 1   x  correspond to the coils a x  to d x  shown in  FIG. 3 , and are connected in series in the same way, and the four outer coils a 2   x  to d 2   x  are also connected in series in the manner shown in  FIG. 3 . Each outer coil a 2   x  to d 2   x  is located on the same radial line as a corresponding one of the inner coils a 1   x  to d 1   x  and all eight coils have substantially parallel axes and substantially the same area-turns product. 
   Let V 1x  be the pickup by the inner set of coils from the interfering source at distance r I  which is given by equation (24) with r=r 1 . 
   
     
       
         
           
             
               
                 
                   V 
                   
                     1 
                     ⁢ 
                     x 
                   
                 
                 = 
                 
                   
                     6 
                     ⁢ 
                     
                       r 
                       1 
                       2 
                     
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           I 
                           4 
                         
                         - 
                         
                           r 
                           1 
                           4 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
   
   Let V 2x  be the pickup by the outer set of coils from the same interfering source. 
   
     
       
         
           
             
               
                 
                   V 
                   
                     2 
                     ⁢ 
                     x 
                   
                 
                 = 
                 
                   
                     6 
                     ⁢ 
                     
                       r 
                       2 
                       2 
                     
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           I 
                           4 
                         
                         - 
                         
                           r 
                           2 
                           4 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
   
   In order to reduce the pickup from this interfering source a fraction, r1 2 /r2 2 , of the outer voltage is subtracted from the inner voltage to give Vx, where 
   
     
       
         
           
             
               
                 
                   V 
                   x 
                 
                 = 
                 
                   
                     V 
                     
                       1 
                       ⁢ 
                       x 
                     
                   
                   - 
                   
                     
                       
                         r 
                         1 
                         2 
                       
                       
                         r 
                         2 
                         2 
                       
                     
                     ⁢ 
                     
                       V 
                       
                         2 
                         ⁢ 
                         x 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
   
   Substituting for V 1x  and V 2x  from equations (26) and (27) gives 
   
     
       
         
           Vx 
           = 
           
             
               
                 6 
                 ⁢ 
                 
                   r 
                   1 
                   2 
                 
                 ⁢ 
                 KdI 
               
               
                 π 
                 ⁡ 
                 
                   ( 
                   
                     
                       r 
                       I 
                       4 
                     
                     - 
                     
                       r 
                       1 
                       4 
                     
                   
                   ) 
                 
               
             
             - 
             
               
                 
                   r 
                   1 
                   2 
                 
                 
                   r 
                   2 
                   2 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     6 
                     ⁢ 
                     
                       r 
                       2 
                       2 
                     
                     ⁢ 
                     KdI 
                   
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         
                           r 
                           I 
                           4 
                         
                         - 
                         
                           r 
                           2 
                           4 
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
   
   Simplifying gives 
   
     
       
         
           
             
               
                 Vx 
                 = 
                 
                   
                     
                       6 
                       ⁢ 
                       KdI 
                     
                     π 
                   
                   ⁢ 
                   
                     
                       r 
                       1 
                       2 
                     
                     
                       r 
                       I 
                       8 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           r 
                           1 
                           4 
                         
                         - 
                         
                           r 
                           2 
                           4 
                         
                       
                       ) 
                     
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               r 
                               1 
                               4 
                             
                             
                               r 
                               I 
                               4 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               r 
                               2 
                               4 
                             
                             
                               r 
                               I 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
   
   Since this is the pickup from the interfering source at distance r I , call this voltage V xB . 
   
     
       
         
           
             
               
                 
                   i 
                   . 
                   e 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     xB 
                   
                 
                 = 
                 
                   
                     
                       6 
                       ⁢ 
                       KdI 
                     
                     π 
                   
                   ⁢ 
                   
                     
                       r 
                       1 
                       2 
                     
                     
                       r 
                       I 
                       8 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           r 
                           1 
                           4 
                         
                         - 
                         
                           r 
                           2 
                           4 
                         
                       
                       ) 
                     
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               r 
                               1 
                               4 
                             
                             
                               r 
                               I 
                               4 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               r 
                               2 
                               4 
                             
                             
                               r 
                               I 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
   
   Consider now the pickup from the same current source when it is located in the measurement position (in the centre of the coil system) when the total voltage pickup is again computed from equation (28). 
   The pickup voltage V 1x  of the inner coil set is given by equation (22) with r=r 1 . 
   
     
       
         
           
             V 
             
               1 
               ⁢ 
               x 
             
           
           = 
           
             
               2 
               ⁢ 
               KdI 
             
             
               π 
               ⁢ 
               
                   
               
               ⁢ 
               
                 r 
                 1 
                 2 
               
             
           
         
       
     
   
   Similarly, the pickup voltage V 2x  of the outer set is given by equation (22) with r=r2. 
   
     
       
         
           
             V 
             
               2 
               ⁢ 
               x 
             
           
           = 
           
             
               2 
               ⁢ 
               KdI 
             
             
               π 
               ⁢ 
               
                   
               
               ⁢ 
               
                 r 
                 2 
                 2 
               
             
           
         
       
     
   
   The total pickup voltage V xa  with the cable in the measurement position is obtained by substituting these values of V 1x  into equation (28) 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         xA 
                       
                       = 
                       
                         
                           
                             2 
                             ⁢ 
                             KdI 
                           
                           
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               1 
                               2 
                             
                           
                         
                         - 
                         
                           
                             
                               r 
                               1 
                               2 
                             
                             
                               r 
                               I 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               KdI 
                             
                             
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 2 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         V 
                         xA 
                       
                       = 
                       
                         
                           
                             2 
                             ⁢ 
                             KdI 
                           
                           π 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 r 
                                 2 
                                 4 
                               
                               - 
                               
                                 r 
                                 1 
                                 4 
                               
                             
                             ) 
                           
                           
                             
                               r 
                               1 
                               2 
                             
                             ⁢ 
                             
                               r 
                               2 
                               4 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
   
   The interference ratio S, given by equation (23) for this new apparatus with inner and outer sets of coils, is 
   
     
       
         
           S 
           = 
           
             
               VxB 
               VxA 
             
             = 
             
               
                 
                   6 
                   ⁢ 
                   
                     
                       KdIr 
                       ⁢ 
                       
                           
                       
                     
                     1 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         r 
                         1 
                         4 
                       
                       - 
                       
                         r 
                         2 
                         4 
                       
                     
                     ) 
                   
                 
                 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       r 
                       I 
                       8 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             r 
                             1 
                             4 
                           
                           
                             r 
                             I 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           r 
                           2 
                           4 
                         
                         
                           r 
                           I 
                           4 
                         
                       
                     
                     ) 
                   
                 
               
               × 
               
                 
                   
                     r 
                     1 
                     2 
                   
                   ⁢ 
                   
                     r 
                     2 
                     4 
                   
                   ⁢ 
                   π 
                 
                 
                   
                     ( 
                     
                       
                         r 
                         2 
                         4 
                       
                       - 
                       
                         r 
                         1 
                         4 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                   ⁢ 
                   KdI 
                 
               
             
           
         
       
     
   
   Simplifying gives 
   
     
       
         
           
             
               
                 S 
                 = 
                 
                   
                     3 
                     ⁢ 
                     
                       r 
                       1 
                       4 
                     
                     ⁢ 
                     
                       r 
                       2 
                       4 
                     
                   
                   
                     
                       
                         r 
                         I 
                         8 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               r 
                               1 
                               4 
                             
                             
                               r 
                               I 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             r 
                             2 
                             4 
                           
                           
                             r 
                             I 
                             4 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
   
   If the inner set on its own had only been used, the interference ratio for that arrangement was given previously by equation (25) with r=r 1  giving 
   
     
       
         
           
             
               
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         r 
                         1 
                       
                       ⁢ 
                       only 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     3 
                     ⁢ 
                     
                       r 
                       1 
                       4 
                     
                   
                   
                     
                       r 
                       I 
                       4 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             r 
                             1 
                             4 
                           
                           
                             r 
                             I 
                             4 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
   
   The interference ratio S of the new apparatus as given by equation (32) is smaller than that for the inner set on its own, as given by equation (33), by the factor 
   
     
       
         
           
             
               r 
               2 
               4 
             
             
               r 
               I 
               4 
             
           
           ⁢ 
           
             1 
             
               ( 
               
                 1 
                 - 
                 
                   
                     r 
                     2 
                     4 
                   
                   
                     r 
                     I 
                     4 
                   
                 
               
               ) 
             
           
         
       
     
   
   For example when r=2r 2  this factor is 0.067, causing a reduction in interference pickup by a factor of 16 approximately. Thus, the worst interference drops from 20% for the inner set on its own to 1.25% when the inner and outer voltages are subtracted in the ratio given by equation (28) i.e. r 1   2 /r 2   2 . When the interfering sources are further, away the reduction factor is even larger. The subtraction of the factor r 1   2 /r 2   2  of the outer voltage from the inner voltage may be implemented with a resistor divider network or as part of an amplifier input stage as will be described with reference to  FIG. 7 . 
   The same considerations apply to the coils detecting the Hy component of the magnetic field. 
     FIG. 6  is a plan view of an embodiment of the invention incorporating interference suppression as described above, and  FIG. 7  is its circuit diagram. 
   In  FIG. 6 , the four inner coils a 1   x  to d 1   x  are mounted on a “C-shaped” insulating motherboard  20  at 90° intervals around the circumference of a notional circle of radius r 1 , and the four outer coils a 2   x  to d 2   x  are mounted on the motherboard  20  at 90° intervals around the circumference of a notional circle of radius r 2 , the two circles being concentric. All eight coils a 1   x  to d 1   x  and a 2   x  to d 2   x  have substantially the same area-turns product and substantially parallel axes and are located substantially in a common plane. 
   Also mounted on the motherboard  20  are eight further coils, an inner set of coils a 1   y  to d 1   y  and an outer set of coils a 2   y  to d 2   y . The coils a 1   y  to d 1   y  and a 2   y  to d 2   y  have substantially parallel axes and substantially the same turns-area product as the coils a 1   x  to d 1   x  and a 2   x  to d 2   x . However, their axes are normal to the axes of the coils a 1   x  to d 1   x  and a 2   x  to d 2   x . Thus each coil a 1   x  to d 1   x  and a 2   x  to d 2   x  forms an orthogonal pair of coils with a corresponding one of the coils a 1   y  to d 1   y  and a 2   y  to d 2   y , wherein in each orthogonal pair the two coils, e.g. the pair of coils a 1   x  and a 1   y , are at substantially the same location on the motherboard  20 , insofar as that is physically practical using the chosen technology, but the axis of one of the coils is rotated through 90° relative to the other coil so that the axis of one coil is tangential to the notional circle on which it lies while the axis of the other coil extends radially of the same circle. Thus the motherboard  20  bears eight orthogonal pairs of coils, four inner pairs a 1   x /a 1   y , b 1   x /b 1   y , c 1   x /c 1   y  and d 1   x /d 1   y  and four outer pairs a 2   x /a 2   y , b 2   x /b 2   y ,  2   x /c 2   y  and d 2   x /d 2   y . As mentioned above, this close positioning of the pairs of coils a 1   x /a 1   y , b 1   x /b 1   y  . . . etc. at the substantially same physical location can be achieved using orthogonal pairs of coils as shown in FIG. 9 of U.S. Pat. No. 5,652,506. 
   The gap  22  in the C-shaped motherboard  20  allows a cable  10 , not shown in  FIG. 6 , to be introduced into the support so as to be positioned at the point P at the centre of the circles of radius r 1  and r 2 , the cable extending normal to the plane containing the coils (i.e. normal to the plane of  FIG. 6 ). In practice the motherboard  20  and coils mounted thereon will be accommodated in a housing (not shown) which could incorporate a clamp or other mechanical device to locate the cable at the point P. Clearly, each of the coils a 1   x  to d 1   x  and a 2   x  to d 2   x  is orientated to pick up the x component Hx of the magnetic field generated by the cable  10 , while each of the coils a 1   y  to d 1   y  and a 2   y  to d 2   y  is orientated to pick up the y component Hy of the magnetic field generated by the cable  10 . 
   The coils are connected as shown in  FIG. 7 :
         Coils a 1   x  and b 1   x  are connected in phase with one another and coils c 1   x  and d 1   x , although connected in phase with one another, are connected in anti-phase to coils a 1   x  and b 1   x  to give an overall output voltage V 1x  at the input to a resistor R 1x .   Coils a 2   x  and b 2   x  are connected in phase with one another and coils c 2   x  and d 2   x , although connected in phase with one another, are connected in anti-phase to coils a 2   x  and b 2   x  to give an overall output voltage V 2x  at the input to a resistor R 2x  (the voltage V 2x  is shown minus because the entire series connection of coils a 2   x  to d 2   x  is connected in reverse polarity to coils a 1   x  to d 1   x ).   Coils a 1   y  and b 1   y  are connected in phase with one another and coils c 1   y  and d 1   y , although connected in phase with one another, are connected in anti-phase to coils a 1   y  and b 1   y  to give an overall output voltage V 1y  at the input to a resistor R 1y .   Coils a 2   y  and b 2   y  are connected in phase with one another and coils c 2   y  and d 2   y , although connected in phase with one another, are connected in anti-phase to coils a 2   y  and b 2   y  to give an overall output voltage V 2y  at the input to a resistor R 2y  (again, the voltage V 2y  is shown minus because the entire series connection of coils a 2   y  to d 2   y  is connected in reverse polarity to coils a 1   y  to d 1   y ).       

   The resistors R 1x  and R 2x  are connected in common to the negative input to an amplifier AMP 1  and are chosen such that: 
   
     
       
         
           
             
               R 
               
                 1 
                 ⁢ 
                 x 
               
             
             
               R 
               
                 2 
                 ⁢ 
                 x 
               
             
           
           = 
           
             ( 
             
               
                 r 
                 1 
                 2 
               
               
                 r 
                 2 
                 2 
               
             
             ) 
           
         
       
     
   
   V out x is therefore given as 
   
     
       
         
           
             
               V 
               out 
             
             ⁢ 
             x 
           
           = 
           
             
               
                 - 
                 R 
               
               
                 1 
                 + 
                 
                   jω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   cR 
                 
               
             
             ⁢ 
             
               ( 
               
                 
                   
                     V 
                     
                       1 
                       ⁢ 
                       x 
                     
                   
                   
                     R 
                     
                       1 
                       ⁢ 
                       x 
                     
                   
                 
                 - 
                 
                   
                     V 
                     
                       2 
                       ⁢ 
                       x 
                     
                   
                   
                     R 
                     
                       2 
                       ⁢ 
                       x 
                     
                   
                 
               
               ) 
             
           
         
       
     
   
   For the frequency range of interest, jωRc is much greater than 1, therefore 
   
     
       
         
           
             
               
                 
                   
                     V 
                     out 
                   
                   ⁢ 
                   x 
                 
                 = 
                 
                   
                     
                       - 
                       1 
                     
                     
                       jω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       c 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           V 
                           
                             1 
                             ⁢ 
                             x 
                           
                         
                         
                           R 
                           
                             1 
                             ⁢ 
                             x 
                           
                         
                       
                       - 
                       
                         
                           V 
                           
                             2 
                             ⁢ 
                             x 
                           
                         
                         
                           R 
                           
                             2 
                             ⁢ 
                             x 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     j 
                     
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cR 
                         
                           1 
                           ⁢ 
                           x 
                         
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         V 
                         
                           1 
                           ⁢ 
                           x 
                         
                       
                       - 
                       
                         
                           
                             R 
                             
                               1 
                               ⁢ 
                               x 
                             
                           
                           ⁢ 
                           
                             V 
                             
                               2 
                               ⁢ 
                               x 
                             
                           
                         
                         
                           R 
                           
                             2 
                             ⁢ 
                             x 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
   
   Similarly, the resistors R 1y  and R 2y  are connected in common to the negative input to an amplifier AMP 2  and are chosen such that: 
   
     
       
         
           
             
               R 
               
                 1 
                 ⁢ 
                 y 
               
             
             
               R 
               
                 2 
                 ⁢ 
                 y 
               
             
           
           = 
           
             ( 
             
               
                 r 
                 1 
                 2 
               
               
                 r 
                 2 
                 2 
               
             
             ) 
           
         
       
     
   
   V out y is therefore given as 
   
     
       
         
           
             
               V 
               out 
             
             ⁢ 
             y 
           
           = 
           
             
               
                 - 
                 R 
               
               
                 1 
                 + 
                 
                   jω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   cR 
                 
               
             
             ⁢ 
             
               ( 
               
                 
                   
                     V 
                     
                       1 
                       ⁢ 
                       y 
                     
                   
                   
                     R 
                     
                       1 
                       ⁢ 
                       y 
                     
                   
                 
                 - 
                 
                   
                     V 
                     
                       2 
                       ⁢ 
                       y 
                     
                   
                   
                     R 
                     
                       2 
                       ⁢ 
                       y 
                     
                   
                 
               
               ) 
             
           
         
       
     
   
   For the frequency range of interest, jωRc is much greater than 1, therefore 
   
     
       
         
           
             
               
                 
                   
                     V 
                     out 
                   
                   ⁢ 
                   y 
                 
                 = 
                 
                   
                     
                       - 
                       1 
                     
                     
                       jω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       c 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           V 
                           
                             1 
                             ⁢ 
                             y 
                           
                         
                         
                           R 
                           
                             1 
                             ⁢ 
                             y 
                           
                         
                       
                       - 
                       
                         
                           V 
                           
                             2 
                             ⁢ 
                             y 
                           
                         
                         
                           R 
                           
                             2 
                             ⁢ 
                             y 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     j 
                     
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                         
                           1 
                           ⁢ 
                           y 
                         
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         V 
                         
                           1 
                           ⁢ 
                           y 
                         
                       
                       - 
                       
                         
                           
                             R 
                             
                               1 
                               ⁢ 
                               y 
                             
                           
                           ⁢ 
                           
                             V 
                             
                               2 
                               ⁢ 
                               y 
                             
                           
                         
                         
                           R 
                           
                             2 
                             ⁢ 
                             y 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
   
   Finally, the current flowing in the cable is calculated in a processor  30  by evaluating
 
 V   out =√{square root over (( V   out   x ) 2 +( V   out   y ) 2 )}{square root over (( V   out   x ) 2 +( V   out   y ) 2 )}
 
and the measured current displayed on a display device such as an LCD panel (not shown). The connections between the various coils can be effected by using conductive tracks (not shown) laid down on the motherboard  20  using printed circuit board (PCB) technology. The amplifiers AMP 1  and AMP 2 , as well as the processor  30 , can be formed by integrated circuit technology and the IC chips located on the motherboard  20  or elsewhere in the device housing.
 
   A total of 16 coils are used in this embodiment and ideally the tuns-area product of these coils should be the same to within 1% at least to obtain accurate results. The cost of 16 wirewound coils accurate to this tolerance could be too expensive for many applications. Planar magnetic printed circuit board coils are a lot cheaper and more accurate to manufacture. However, for a true implementation of the apparatus shown in  FIGS. 6 and 7 , in each pair of orthogonal coils the geometric centre of each of the coils a 1   x , b 1   x , etc. picking up the Hx component should be located at the same position as the geometric centre of the corresponding coil a 1   y , b 1   y , etc. picking up the Hy component. This is possible using the technique shown in FIG. 9 of U.S. Pat. No. 5,652,506 but not with planar magnetic coils as the tracks are confined to one plane and it is not possible with present day techniques to simultaneously have tracks on orthogonal planes in the same PCB. However, a slight compromise with planar magnetic printed circuit board coils works very well and this arrangement is shown in  FIGS. 8(   a ) to  8 ( c ).  FIG. 8  actually shows the PCB implementation of the two orthogonal coil pairs a 1   x /a 1   y  and a 2   x /a 2   y ) but the same principle is applicable to the two orthogonal coil pairs on the same radius at each of the other three quadrants of the motherboard  20 . 
   The coils a 1   y  and a 2   y  are substantially identical and each is formed as a conductive track  40  on an insulating substrate  42 . Although only one side of the substrate is seen in  FIG. 8(   c ), tracks  40  are formed on each opposite surface of the substrate and connected in series through a central via hole  44 . Both tracks  40  form the coil whose opposite ends are connected to respective solder pads  46  formed on tabs extending down from the main body of the substrate  42 . 
   By contrast, each coil a 1   x  and a 2   x  is formed in two parts. Considering coil a 1   x , it is formed in two parts a 1   x ( 1 ) and a 1   x ( 2 ). The parts a 1   x ( 1 ) and a 1   x ( 2 ) are formed as conductive tracks  50  on respective insulating substrates  52 . However, each of the parts a 1   x ( 1 ) and a 1   x ( 2 ) has a turns-area product half that of the coil a 1   y . This can be achieved by providing double the number of turns on the coil a 1   y  than the number on parts a 1   x ( 1 ) and a 1   x ( 2 ). 
   Similarly, the coil a 2   x  is formed in two parts a 2   x ( 1 ) and a 2   x ( 2 ), again formed as conductive tracks  50  on respective insulating substrates  52  and each having a turns-area product half that of the coil a 2   y . Actually, in this embodiment the parts a 1   x ( 1 ) and a 2   x ( 1 ) are formed on one common substrate  52  and likewise the parts a 1   x ( 2 ) and a 2   x ( 2 ) are formed on another common substrate  52 , but this is not necessary. 
   The substrates  42 ,  52  are mounted upstanding vertically in the motherboard  20  by inserting the solder tabs  46 ,  56  into slots in the motherboard and soldered to tracks on the motherboard. The arrangement is as shown in  FIG. 8(   a ). The coil a 1   y  is embraced on each side by the coils a 1   x ( 1 ) and a 1   x ( 2 ) normal thereto, and the coil a 2   y  is embraced on each side by the coils a 2   x ( 1 ) and d 2   x ( 2 ) normal thereto. The solder tabs  56  are connected by conductive tracks on the motherboard  20  to connect the coils a 1   x ( 1 ) and a 1   x ( 2 ) in series in phase to form the coil a 1   x  and the coils a 2   x ( 1 ) and a 2   x ( 2 ) in series in phase to form the coil a 2   x . Since the coil parts a 1   x ( 1 ) and a 1   x ( 2 ) combined have the same turns-area product as the coil a 1   y  and are equally spaced at either end of coil a 1   y , as a pair they have the same geometrical centre as coil a 1   y . Similarly, as a pair the coil parts a 2   x ( 1 ) and a 2   x ( 2 ) they have the same geometrical centre as coil a 2   y . The remaining connections are as shown in  FIG. 7 . 
   Modifications of the above embodiment are possible. For example, the turns-area product of the coils a 2   x  to d 2   x  could be different to that of the coils a 1   x  to d 1   x , provided allowance is made for this in the relative values of the resistors R 1x  and R 2x  or elsewhere in the circuit. Similarly, the turns-area product of the coils a 2   y  to d 2   y  could be different to that of the coils a 1   y  to d 1   y  provided suitable allowance is made elsewhere. Also, if interference from external sources is not probable in the circumstances likely to be encountered in use, the outer sets of coils, i.e. the orthogonal pairs of coils located on the circle of radius r 2  in  FIG. 6 , can be omitted. Further, since diametrically opposite sets of coils are provided primarily to provide a larger signal and to further reduce external interference, as well as reducing errors due to movement of the cable from centre point P, the invention could be implemented with just two sets of coils at 90° spacing, e.g. the sets of coils at the 3 o&#39;clock and 6 o&#39;clock positions of  FIG. 6 . 
   The invention is not limited to the embodiments described herein which may be modified or varied without departing from the scope of the invention.