Abstract:
A method and apparatus for simultaneously measuring voltage and current in a primary high voltage conductor. A current and a voltage from the primary conductor is monitored. The current and voltage information present on the current transformer secondary winding is separated. The voltage information present on the current transformer secondary winding is used to provide a voltage measurement output proportional to the voltage present on the primary conductor. The current information present on the current transformer secondary winding is used to provide a current measurement output proportional to the current flowing on the high voltage conductor.

Description:
TECHNICAL FIELD 
     This invention relates to voltage and current sensing. 
     BACKGROUND 
     Capacitively-coupled voltage measurement is frequently used to measure the voltage present on a high voltage conductor in high voltage alternating current electric systems. Typically, a high voltage capacitor is connected between the high voltage conductor and the secondary winding, and a load capacitor is connected between the secondary winding and the toroidal ferro-magnetic core. The high voltage capacitor and the load capacitor form a simple capacitive voltage divider from which the voltage of the high voltage conductor may be determined. Voltage measurement is often supplemented with a measurement of current flowing through the high voltage conductor. Typically, a current transformer is used to provide this current measurement by surrounding the high voltage conductor with a ferro-magnetic transformer core around which an insulated secondary winding is wound uniformly. 
     Although capacitively-coupled voltage sensing is widely used, the cost and precision of the capacitively-coupled sensors are closely related to the quality of the high voltage capacitors used to perform the measurements. High precision is often achieved by using closely matched foil capacitors immersed in a dielectric liquid or ceramic capacitors built with high-performance, temperature-compensating materials. These high precision capacitors generally are quite expensive. 
     A low cost approach is achieved by constructing a voltage-sensing capacitor as an integral part of the high voltage apparatus. The capacitance of such a capacitor is determined by the internal device geometry and the dielectric constant of an associated insulating material. The low cost approach often produces a relatively low capacitance value that limits the overall measurement accuracy of the design. Low capacitance, and therefore low energy, also presents a challenge in transmitting the measured information from the sensor to the device that is performing the voltage measurement. 
     Parasitic capacitance between the current transformer secondary winding and the high voltage conductor may elevate the potential of the secondary winding, which may lead to failure of the secondary winding insulation. A similar problem applies to the ferro-magnetic based transformer core if the potential is left freely floating with respect to the high voltage conductor potential. To reduce or eliminate this current transformer failure mechanism, the standard approach has been to ground the current transformer core or to add a grounded shielding electrode that protects the current transformer secondary winding. 
     SUMMARY 
     In one general aspect, simultaneous measurements of voltage and current present on a primary high voltage conductor are achieved through use of a current transformer. Capacitive coupling between the high voltage conductor and the current transformer secondary winding is used to transmit primary voltage information without affecting the current normally flowing through the current transformer secondary winding. The voltage and current information is separated, the voltage information is used to provide a voltage measurement output representative of the voltage present on the primary conductor, and the current information is used to provide a current measurement output representative of the current flowing on the high voltage conductor. The same approach may be used for each phase in a multi-phase system. Thus, capacitively coupled voltage measurement may be combined with current measurement in a single device. The approach exploits the parasitic capacitance normally present between the high voltage conductor and the current transformer secondary winding, and the parasitic capacitance normally present between the current transformer secondary winding and the ferro-magnetic core to form a simplified capacitive voltage divider from which the voltage of the high voltage conductor may be determined. 
     Implementations may include one or more of the following features. For example, the current measurement may be obtained through an electronic circuit or a secondary transformer, and the voltage measurement may be obtained through an electronic circuit. A neutral current measurement may also be obtained, for example, through an electronic circuit or a transformer. 
     A capacitive voltage divider may be used in providing the voltage measurement. In multi-phase systems, each phase would have a corresponding capacitive voltage divider. The capacitive voltage divider may have first and second capacitances, where the first capacitance is between the high voltage conductor and the secondary winding of the current transformer and the second capacitance is between the secondary winding of the current transformer and the transformer core. The first and second capacitance may be, for example, the parasitic capacitance between the high voltage conductor and the secondary winding of the current transformer and the parasitic capacitance between the secondary winding of the current transformer and the transformer core of the current transformer connected to a reference potential. A ground potential may be used as a reference potential. In another implementation, any potential may be used as the reference potential. For example, any potential with a voltage difference with respect to the high voltage conductor being measured may be used. In one implementation, the second capacitance may be increased by adding an external capacitor between the current transformer secondary winding and the reference potential. In another implementation, the second capacitance may be increased by adjusting the parasitic capacitance. The parasitic capacitance may be adjusted, for example, by adjusting the device geometry. In one implementation, the second capacitance has a value from approximately 0.001 to 10 microfarads and forms a high pass filter network in combination with a drain resistor that is connected between the current transformer secondary winding and ground, where the cutoff frequency of the high pass filter network may be set between approximately 1 to 0.001 hertz. 
     An electronic circuit may be used in providing the voltage measurement. For example, the electronic circuit may have an operational amplifier, a resistor connected to the operational amplifier and a terminal of the current transformer, and a drain resistor connected to the operational amplifier. 
     In another implementation, an auxiliary transformer may be used to provide the current measurement. Alternatively, an electronic circuit may be used to provide the current measurement. The electronic circuit may include an operational amplifier connected to the current transformer and a burden resistor connected to the operational amplifier. 
     In a further implementation, the neutral current of a multi-phase system may be measured. For example, a transformer with a separate winding for each phase may be used to provide the neutral current measurement. Alternatively, the neutral current may be measured using an electronic circuit. 
     Another implementation includes canceling from the voltage measurement crosstalk induced by one or more additional phases in a multi-phase system. For example, voltage measurements may be obtained for the additional phases, a product may be generated for each additional phase by multiplying the additional phase voltages by a corresponding predetermined constant, and the product for each additional phase may be subtracted from the voltage measurement. In one example, there are three phases in the multi-phase system. 
     The crosstalk may be cancelled by an electronic circuit. The electronic circuit may include an operational amplifier, a connecting resistor connected between the input and output of the operational amplifier, and a resistor associated with an additional phase connected to the operational amplifier. In another implementation, the crosstalk may be cancelled by computer software. 
     The current transformer secondary winding may be protected from insulation failure induced by a transient voltage. For example, a surge suppressor may be connected between the transformer secondary winding and ground. 
     Although primarily intended for medium voltage power systems, simultaneous measurement techniques may be applied to other voltage levels and system frequencies. Moreover, by reducing the number of components required, the techniques offer a very low cost solution for combined current and voltage measurement. Components for implementing the approach may be retrofitted to existing systems to add voltage sensing capability to older transformer installations. The techniques may be used in a multi-phase system, such as a three-phase system, or in a single phase system. 
     Other features and advantages will be apparent from the description and drawings, and from the claims. 
    
    
     DESCRIPTION OF DRAWINGS 
     FIG. 1 is a front view of a three-phase auto-recloser system using a voltage sensor with active crosstalk cancellation. 
     FIG. 2 is a partially cut-away side view showing the internal construction of a single module, including a voltage sensor, of the three-phase auto-recloser system shown in FIG.  1 . 
     FIG. 3 is a cross-sectional side view of a combined current and voltage sensor. 
     FIG. 4 is a cross-sectional side view illustrating parasitic capacitance in the sensor of FIG.  3 . 
     FIG. 5 is a block diagram of a current and voltage sensor for single-phase voltage. 
     FIG. 6 is a block diagram of a current and voltage sensor for three-phase voltage. 
     FIG. 7 is a block diagram of a voltage sensor and crosstalk cancellation system used by the system of FIG.  1 . 
     FIG. 8 is a schematic diagram of an electronic circuit for performing crosstalk cancellation used by the cancellation system of FIG.  7 . 
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     FIG. 1 shows a three-phase auto-recloser  100  that is connected by a signal transmission cable  105  to an electronic control  110 . High voltage conductors (not shown) are connected to terminals  115 ,  120 ,  125 ,  130 ,  135 , and  140 , extending from modules  145  of the auto-recloser  100 . 
     Referring to FIG. 2, each module  145  includes a capacitively-coupled voltage sensor  200  integrated around a side arm conductor  205  associated with a respective one of terminals  115 ,  120 , and  125 . The voltage sensing electrode  200  is axially symmetric about the high voltage conductor  205  and placed within the diameter of a current transformer  210 . A current transformer corona shield  215  surrounds the current transformer and provides additional dielectric shielding for the voltage sensing electrode  200  to reduce external field effects. The combination of the capacitively-coupled voltage sensor and the current transformer is used to produce the voltage and current measurements. 
     As illustrated in FIG. 3, the combined current and voltage sensor  200  can be positioned within a solid insulating body  300  or immersed into an insulating gas or liquid. The sensor  200  includes a toroidal ferro-magnetic core  305  and an insulated secondary winding  310  that is wound uniformly around the core  305 . The current carrying high voltage conductor  315  is centered and passes through the central core opening. FIG. 3 shows the combined current and voltage sensor  200  for a single phase. In a multi-phase power system network, a sensor  200  is needed for each individual phase. 
     FIG. 4 shows the parasitic capacitance present in the combined current and voltage sensor system. Specifically, FIG. 4 shows the parasitic capacitance  400  present between the secondary winding  405  and the high voltage conductor  410 . In addition, FIG. 4 shows the parasitic capacitance  415  present between the secondary winding  405  and the transformer core  420 . The transformer core is typically connected to a reference potential, which may be electrical ground or another potential. Once transformer core  420  is grounded or placed at the reference potential, parasitic capacitances  400  and  415  form a simple capacitive divider. In this configuration, the secondary winding voltage will float at the output potential determined by the following equation:          V   out     =       V     i                 n       ×         C   1         C   1     +     C   2         .                              
     In the equation, C 1  is the parasitic capacitance  400  between the high voltage conductor and the secondary winding and C 2  is the parasitic capacitance  415  between the secondary winding and the transformer core. 
     The value of capacitor  400  is determined by the design of the current transformer. The output potential V out  can be adjusted by increasing the value of the capacitance  415  so as to adjust the voltage divider ratio. The value of the capacitance  415  may be increased, for example, by adding an external capacitor between the current transformer secondary winding and the reference potential and/or by adjusting the value of the parasitic capacitance present between the secondary winding and the transformer core. The measured voltage and current signals are combined on a single pair of conductors (i.e., the current transformer secondary wires) and must be separated for actual measurement and display. 
     FIG. 5 shows a circuit that separates the voltage and current signals using a differential amplifier in combination with a burden resistor. The output voltage level can be adjusted by varying the value of capacitance  415  (shown in FIG. 4) in current transformer  505 . The value of the capacitance  415  may be increased, for example, by adding an external capacitor between the current transformer secondary winding and the reference potential and/or by adjusting the value of the parasitic capacitance present between the secondary winding and the transformer core. For example, the value of capacitors  540  and/or  550  may be varied. Under normal operating conditions, the output voltage V out  typically is set between 0.5 and 10 V RMS . Surge protection components may be introduced into the circuit to limit the maximum voltage that can be developed during power system transients, lightning strikes, and other over-voltage events. The surge suppressor protective level is normally coordinated at approximately 110% to 500% of the typical steady state operating level. Different surge suppressor technologies such as MOV, TVS, Sidactor, and Sparc-Gap may be used. 
     High voltage conductor  501  carries a current I and a voltage V, and is coupled to current transformer  505 . Current transformer  505  is connected to a voltage measuring circuit  510  and a current measuring circuit  515 . 
     The voltage measuring circuit  510  includes a capacitor  540  and a surge protection component  545  that are connected in parallel between a terminal  516  of current transformer  505  and ground. A resistor  520  is connected between a terminal  516  of current transformer  505  and an input  533  to an operational amplifier  535 . The other input  534  to operational amplifier  535  is connected to ground. A capacitor  550  and a surge protection component  555  are connected in parallel between a terminal  517  of current transformer  505  and ground. A resistor  525  is connected between the terminal  517  of current transformer  505  and the input  533  to operational amplifier  535 . A drain resistor  530  is connected between the input terminals  533  and  534  of operational amplifier  535 . The output  536  of operational amplifier  535  is proportional to the voltage of high voltage conductor  501 . 
     The current measuring circuit  515  includes a burden resistor  560  connected between terminal  516  and terminal  517  of current transformer  505 . The burden resistor  560  is further connected between input terminals  563  and  564  of an operational amplifier  565 . The output  566  of operational amplifier  565  is proportional to the current in high voltage conductor  501 . In other implementations, the described operational amplifier and burden resistor combination are replaced with an auxiliary transformer. 
     FIG. 6 shows a circuit  600  to extract the zero sequence (neutral) current  690  information in the case of a multi-phase power system network. This neutral current information is often necessary in a multi-phase power system network. The neutral current is extracted by summing together the three individual phase currents. The circuit also provides outputs for the individual phase voltages  636   A ,  636   B , and  636   C , and outputs for the individual phase currents  666   A ,  666   B , and  666   C . 
     High voltage conductors  601   A ,  601   B , and  601   C  carry currents I A , I B , and I C  and voltages V A , V B , and V C , and are coupled to current transformers  605   A ,  605   B , and  605   C  respectively. Each of current transformers  605   A ,  605   B  and  605   C  is connected to a corresponding one of voltage measuring circuits  610   A ,  610   B , and  610   C , and to a corresponding one of current measuring circuits  615   A ,  615   B , and  615   C . Current transformers  605   A ,  605   B , and  605   C  are further connected to neutral current measuring circuit  695 . 
     For ease of description, components of the voltage measuring circuits  610   A ,  610   B , and  610   C  and the current measuring circuits  615   A ,  615   B , and  615   C  are referred to collectively rather than individually. Thus, for example, capacitors  640   A ,  640   B , and  640   C  are referred to as capacitor  640 . 
     Each voltage measuring circuit  610  includes a capacitor  640  and a surge protection component  645  that are connected in parallel between a terminal  616  of current transformer  605  and ground. A resistor  620  is connected between a terminal  616  of current transformer  605  and an input  633  to an operational amplifier  635 . The other input  634  to operational amplifier  635  is connected to ground. A capacitor  650  and a surge protection component  655  are connected in parallel between a terminal  617  of current transformer  605  and ground. A resistor  625  is connected between a terminal  617  of current transformer  605  and an input  633  to operational amplifier  635 . A drain resistor  630  is connected between the input terminals  633  and  634  of operational amplifier  635 . The output  636  of operational amplifier  635  is proportional to the voltage of the corresponding high voltage conductor  601 . 
     Each of the current measuring circuits  615  includes an auxiliary transformer  665  connected between terminal  616  and terminal  617  of current transformer  605 . The output  666  of auxiliary transformer  665  is proportional to the current in the corresponding high voltage conductor  601 . In other implementations, an operational amplifier and burden resistor combination may be substituted for the described auxiliary transformer  665 . 
     The neutral current measuring circuit  695  includes windings  670   A ,  670   B , and  670   C  of an auxiliary transformer  680 . These windings are connected between auxiliary transformer  665   A ,  665   B , and  665   C  and current transformer  605   A ,  605   B , and  605   C . The neutral current output  690  sensed by transformer  680  is proportional to the sum of the three phase currents I A , I B , and I C . 
     Referring to FIG. 7, each of phases V A , V B , and V C  of the three-phase AC voltage is measured by an associated one of the high-voltage capacitive sensors  701   A ,  701   B , and  701   C . The outputs of the three high voltage capacitor sensors  701   A ,  701   B , and  701   C  are combined by a signal processing circuit  700  located within a housing of the auto-recloser  100  (FIG.  1 ). The signal processing circuit  700  includes load capacitors  705   A ,  705   B , and  705   C , which are used to form simple capacitive voltage dividers in combination with the high voltage capacitors  701   A ,  701   B , and  701   C . The output of each of the voltage dividers is connected to a corresponding one of surge protection networks  710   A ,  710   B , and  710   C  and drain resistors  715   A ,  715   B , and  715   C . The drain resistors are used to eliminate any static charge which may be present on the outputs of the voltage dividers. 
     The signals then are passed to high impedance buffer circuits  720   A ,  720   B , and  720   C , which are used to minimize the voltage sensor phase error. From the buffer circuits, the signals pass through programmable gain stages  725   A ,  725   B , and  725   C  to account for manufacturing tolerances of the high voltage capacitors  701   A ,  701   B , and  701   C . The programmable gain stages correct the individual sensor ratio so that the divider ratios are the same for each phase of the three-phase AC voltage. The required gain calibration parameters for the programmable gain stages can be programmed by using a calibration port  730  and are stored in non-volatile memory  735 . 
     The calibrated individual sensor outputs are fed through temperature compensation circuits  740   A ,  740   B , and  740   C , which use ambient temperature measurements to compensate for the temperature variations of the individual capacitive voltage dividers. The value required for temperature compensation is determined by the type of dielectric used in capacitors  701  and  705 , and is constant in any given sensor design. 
     Next, the individual sensor outputs are fed through crosstalk compensation circuits  745   A ,  745   B , and  745   C  to provide first order crosstalk cancellation. Alternatively, higher orders of crosstalk cancellation may be provided. The crosstalk cancellation may be performed by signal processing hardware, and may be implemented, for example, as an application-specific integrated circuit (ASIC). Alternatively, the crosstalk cancellation may be performed by a computer program running on either a general purpose computer or a special purpose computer. Crosstalk cancellation minimizes the effect of crosstalk between the three voltage sensors. The crosstalk is caused by the simple high voltage capacitor construction and the relative proximity of the three-phase voltage conductors. 
     In the presence of crosstalk, the measured voltage present on the output of the individual voltage sensors can be described by the following equation (1): 
       V   A   measured   =V   A   +k   1   V   B   +k   2   V   C   
     
       
           V   B   measured   =V   B   +k   3   V   A   +k   4   V   C   (1) 
       
     
     
       
         
           V 
           C 
           measured 
           =V 
           C 
           +k 
           5 
           V 
           A 
           +k 
           6 
           V 
           B 
         
       
     
     The system of equations in (1) is a system of three equations with three unknowns, namely V A , V B , and V C . These unknowns, V A , V B , and V C , are the voltages to be measured. The system of equations above describes a linear superposition caused by the proximity of the three sensors to each other and the imperfect shielding of the individual sensors from crosstalk. The three sensors for phases A, B, and C are used to measure three voltages, V A   measured , V B   measured , and V C   measured . The measured voltage of each phase contains crosstalk terms from the other two phases. For example, the measured voltage V A   measured  contains the term k 1 V B  from phase B and the term k 2 V C  from phase C. In equation (1), the coupling constants k 1 , k 2 , k 3 , k 4 , k 5 , and k 6  are determined by the auto-recloser device geometry. The coupling constants can be measured and will remain constant as long as the device geometry is kept constant. For the symmetric three-phase design shown in FIG. 1, the coupling constants are symmetric (k 1 =k 3 =k 4 =k 6  and k 2 =k 5 ). The full solution of the system of equations in (1) is shown below in equation (2).                  V   A     =               V   A   MEAS     -       k   4          k   6          V   A   MEAS       -       k   1          V   B   MEAS       +                   k   2          k   6          V   B   MEAS       -       k   2          V   C   MEAS       +       k   1          k   4          V   C   MEAS                 1   -       k   1          k   3       -       k   2          k   5       -       k   4          k   6       +       k   1          k   4          k   5       +       k   2          k   3          k   6                  
            V   B     =                 -     k   3            V   A   MEAS       +       k   4          k   5          V   A   MEAS       +     V   B   MEAS     -                   k   2          k   5          V   B   MEAS       -       k   4          V   C   MEAS       +       k   2          k   3          V   c   MEAS                 1   -       k   1          k   3       -       k   2          k   5       -       k   4          k   6       +       k   1          k   4          k   5       +       k   2          k   3          k   6                  
            V   C     =                 -     k   5            V   A   MEAS       +       k   3          k   6          V   A   MEAS       -       k   6          V   B   MEAS       +                   k   1          k   5          V   B   MEAS       +     V   C   MEAS     +       k   1          k   3          V   c   MEAS                 1   -       k   1          k   3       -       k   2          k   5       -       k   4          k   6       +       k   1          k   4          k   5       +       k   2          k   3          k   6                     (   2   )                                
     Equation (2) can be simplified when the crosstalk levels (as indicated by coefficients k 1 , k 2 , k 3 , k 4  k 5 , and k 6 ) are sufficiently low. For example, when the crosstalk levels are equal to or less than approximately 0.1 (10%), the set of equations in (2) may be simplified so that the corrected output values are described by the following equation (3): 
     
       
         
           V 
           A 
           corrected 
           =V 
           A 
           measured 
           −k 
           1 
           V 
           B 
           measured 
           −k 
           2 
           V 
           C 
           measured 
         
       
     
     
       
         
           V 
           B 
           corrected 
           =V 
           B 
           measured 
           −k 
           3 
           V 
           A 
           measured 
           −k 
           4 
           V 
           C 
           measured 
         
       
     
     
       
           V   C   corrected   =V   C   measured   −k   5   V   A   measured   −k   6   V   B   measured   (3 ) 
       
     
     Equation (3) is derived from equation (2) as follows. First, consider the denominator of equation (2). The denominator can be approximated as the value of 1 when the crosstalk levels are sufficiently low, for example 0.1 or less. The second term in the denominator of equation (2) is equal to or less than 0.01 if the coefficients k 1  and k 3  are equal to or less than 0.1 because k 1 k 3 ≦0.1*0.1=0.01. The same analysis applies to the third term, k 2 k 5 , and the fourth term, k 4 k 6 . Therefore, the second, third, and fourth term each contribute 1% error or less. The fifth and sixth terms are even smaller. The fifth term is equal to or less than 0.001 if the coefficients k 1 , k 4 , and k 5  are equal to or less than 0.1 because k 1 k 4 k 5 ≦0.1*0.1*0.1=0.001. The same analysis applies to the sixth term, k 2 k 3 k 6 . Therefore, the fifth and sixth terms each contribute 0.1% error or less. When k 1 , k 2 , k 3 , k 4  k 5 , and k 6  are equal to or less than 0.1, the denominator becomes 1−0.01−0.01−0.01+0.001+0.001=0.972, which is almost equal to 1.00. Thus, for crosstalk terms approximately equal to or less than 0.1, the denominator effectively reduces to 1. 
     The numerator can be simplified in a similar fashion. When the crosstalk levels are sufficiently low, for example 0.1 or less, the second, fourth, and sixth terms in the numerator are small contributors which can be eliminated. For example, in the second term of the first equation in (2), the factor k 4 k 6 ≦0.1*0.1=0.01. Eliminating the small contributors in the numerator of equation (2) results in the simplified first order crosstalk cancellation of equation (3). 
     Equation (3), as simplified from equation (2), only satisfies the first order crosstalk cancellation because the measured terms already contain errors introduced by adjacent sensors in the other phases. However, it is appropriate to use equation (3) in certain cases, such as an analog circuit implementation with crosstalk levels (as indicated by coefficients k 1 , k 2 , k 3 , k 4  k 5 , and k 6 ) approximately equal to or less than 0.1 (10%). 
     Because the values in equation (3) contain second order errors due to the simplification from equation (2), the resulting voltages at the left hand side of equation (2) are not called V A , V B , and V C . Instead, the terms V A   corrected , V B   corrected , and V C   corrected  are used to capture this difference between equations (2) and (3). 
     As previously mentioned, the crosstalk cancellation described above may be performed by signal processing hardware, and may be implemented, for example, as an application-specific integrated circuit (ASIC). Alternatively, the crosstalk cancellation may be performed by a computer program running on either a general purpose computer or a special purpose computer. 
     After crosstalk cancellation is performed, the sensor output for each of phases V A , V B , and V C  of the three-phase AC voltage is fed to a corresponding one of differential output drivers  750   A ,  750   B , and  750   C . The output drivers  750  amplify the measurement signals for each phase V A , V B , and V C  of the three-phase AC voltage and make them ready for transmission through a cable. Differential outputs are used to enhance the immunity of the transmitted signal to externally induced noise. Finally, the sensor outputs are fed to surge protection networks  755   A ,  755   B , and  755   C  for transmission on the cable  105 . 
     Referring to FIG. 8, a circuit  800  for economically performing the crosstalk cancellation function is shown for a single phase, in this case phase “A,” of the three-phase system. The inputs V A , V B , and V C  shown in FIG. 8 may be obtained, for example, from outputs  636   A ,  636   B , and  636   C  of FIG.  6 . The inputs V A , V B , and V C  shown in FIG. 8 are proportional to the voltages of high voltage conductors as shown, for example, by  601   A ,  601   B , and  601   C  of FIG.  6 . 
     Input V A  is connected to the positive input  810  of an operational amplifier  805 . A resistor  825  is connected between input V B  and the negative input  815  of operational amplifier  805 . A resistor  830  is connected between input V C  and the negative input  815  of operational amplifier  805 . Resistor  835  is connected between the negative input  815  and the output  820  of operational amplifier  805 . The output  820  of operational amplifier  805  represents the first order crosstalk cancellation of the errors introduced by phases B and C into the measurement of phase A, as shown in equation (2) above. The same approach just described for one phase applies equally to the other two phases. 
     It will be understood that various modifications may be made. For example, the crosstalk compensating function can be performed in software on a programmable numeric device. Such an implementation is also an attractive way to apply the full solution to a simple system of linear equations shown in equation (3), so as to eliminate higher order errors introduced by equation (2). 
     As another example, in FIG. 5 the differential amplifier/burden resistor combination may be substituted with an auxiliary current transform. Also, in FIG. 5, a single capacitor and/or resistor with center tapped auxiliary current transformer. It is also possible to eliminate resistor R. 
     Accordingly, other implementations are within the scope of the following claims.