the following detailed description and appended drawings describe and illustrate various exemplary embodiments of the invention . the description and drawings serve to enable one skilled in the art to make and use the invention , and are not intended to limit the scope of the invention in any manner . in respect of the methods disclosed , the steps presented are exemplary in nature , and thus , the order of the steps is not necessary or critical . referring now to the drawings , and particularly fig1 , there is shown generally at 2 a portion of a multi - layer printed circuit board . in this embodiment , the printed circuit board 2 includes a first surface layer 12 , a second surface layer 14 , a first inner layer 16 , and a second inner layer 18 stacked together . in the embodiment shown , the layers 12 , 14 , 16 , 18 of the printed circuit board 2 are constructed from an insulative circuit board substrate 10 adapted such that conductive traces may be deposited or etched using known photoresitive processes . an outer coil 20 is wound in a first direction by disposing conductive traces on outer surfaces of the first surface layer 12 and the second surface layer 14 . the traces are connected in a helical manner using a series of conductive plated outer coil via holes 25 that traverse the first surface layer 12 , the second surface layer 14 , the first inner layer 16 , and the second inner layer 18 of the printed circuit board 2 . an inner coil 30 is wound in a second direction by disposing conductive traces on surfaces of the first inner layer 16 and the second inner layer 18 facing the layers 12 and 14 respectively . the inner coil 30 traces are connected in a helical manner using a series of conductive plated inner coil via holes 35 that traverse the first inner layer 16 and the second inner layer 18 . the inner coil 30 windings in the second direction are electrically opposite to that of the first direction of the outer coil 20 windings . additionally , the inner coil 30 is constructed with a higher - turn density than the outer coil 20 in order to compensate for the size difference between the outer coil 20 and inner coil 30 . the outer coil 20 and the inner coil 30 are square - shaped in cross section . it is understood that other windings can be used as desired . an aperture 40 is milled in the printed circuit board 2 as illustrated in fig2 . the aperture 40 is adapted to receive a solid conductive bus bar . the outer coil 20 and the inner coil 30 extend about the periphery of the aperture 40 . alternatively , the aperture 40 may be adapted to receive one or more various types of electrical conductors without departing from the scope of this invention . in fig3 , an integrator circuit 50 is linked to an output 48 . the output 48 includes an electrical connection 46 to the outer coil 20 , and an electrical connection 44 to the inner coil 44 . in operation , the prior art rogowski coil 100 is placed around one or more electrical conductors 115 whose instantaneous current i ( t ) is to be measured , as illustrated in fig4 . the instantaneous current i ( t ) for alternating current waveforms is given by i max sin ( ωt ) where i max is the amplitude of the current . the voltage v ( t ) induced in the rogowski coil 100 is defined by the first equation : where φ j is the instantaneous flux for the j - th turn of the total n turns . if the core has a constant cross section s , the wire is wound perpendicularly to a line 120 that is centered in the cross section s , and the wire is wound with a constant density equal to n , where the mutual reactance m is defined by m = u o ηs , then equation 1 may be written as : according to the above equation the electrical conductor will induce a voltage on the rogowski coil 100 proportional to the rate of change of the measured current i ( t ) in the electrical conductor . the rogowski coil output voltage v ( t ) can be measured at output 48 . the voltage output of the rogowski coil v ( t ) is a function of the time - rate - of - change of the ac current passing within the perimeter of the coils . the integrator circuit 50 is necessary in order to attain the actual waveform of interest which is an output voltage v out that is proportional to the measured current i ( t ). output voltage integration may be performed using a variety of means commonly known in the art . the output voltage signal v out can be measured at the integrator output 52 . as shown in fig1 , there is a difference in size between the outer coil 20 and inner coil 30 . this difference in size of the outer coil 20 and inner coil 30 results in a difference in induced voltage , both by the current of interest i ( t ) and by magnetic fields created by conductors outside the printed circuit board aperture 40 . in order to compensate for this difference in induced voltage , it is desirable that the inner coil 30 is constructed with a higher turns density than the outer coil 20 . the effective electrical geometries of the outer coil 20 and inner coil 20 can be made substantially identical by constructing the inner coil 30 with a higher turns density than the outer coil 20 . the maximum effective turns density is largely determined by the minimum achievable hole size during printed circuit board fabrication , and thicker printed circuit boards impose a larger minimum hole size constraint for plated vias . therefore , the inner coil 30 can be constructed with a higher turns density , because the inner coil vias 35 traverse two substrate 10 layers , while the outer coil vias 25 traverse all four of the substrate 10 layers . the outer coil 20 and inner coil 30 are wound in electrically opposite directions to cancel the undesirable electrical fields from outside of the aperture 40 . this maximizes the accuracy of the measurement capability of the rogowski coil . from the foregoing description , one ordinarily skilled in the art can easily ascertain the essential characteristics of this invention and , without departing from the spirit and scope thereof , make various changes and modifications to the invention to adapt it to various usages and conditions .