Patent Publication Number: US-6984979-B1

Title: Measurement and control of magnetomotive force in current transformers and other magnetic bodies

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   Not Applicable 
   STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
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   REFERENCE TO A MICROFICHE APPENDIX 
   Not Applicable 
   BACKGROUND OF THE INVENTION 
   This invention relates to the measurement and control of magnetomotive forces that influence magnetic fluxes in magnetic bodies. When applied to current transformers, the invention relates to the measurement of electric current, especially electric current that has a d-c (direct-current) component. 
   Magnetomotive force (F) is associated with the art of magnetic circuits, and is often defined for a closed loop as the line integral of the magnetic field strength (H) around the closed loop:
 
 F=     H ·dl  
 
Magnetomotive force F is a scalar quantity associated with the closed loop, while magnetic field strength H is a vector quantity. By Ampere&#39;s Law, magnetomotive force is proportional to the total current flowing through the closed loop. Utilizing the meter-kilogram-second (m.k.s.) system of units, magnetomotive force has units of amperes (or amp-turns), and is equal to the total current flowing through the closed loop. The closed loop is often chosen to pass through one or more conductive windings wrapped around a magnetic body, and a magnetomotive force equal to the current in the winding times the number of winding turns is associated with each winding (thus the unit “amp-turns”). The total magnetomotive force for the closed loop is the sum of all amp-turn contributions from all windings and other conductors.
 
   Many magnetic devices operate best with an average magnetomotive force near zero. A deviation away from zero often results in excessive buildup of magnetic flux that causes the device to malfunction. Ordinary current transformers are one type of device for which this is usually the case. Since an understanding of current transformer operation is important to understanding some embodiments of the invention, the background of current transformers will be discussed in some detail. However, it should be kept in mind that the invention is applicable to many different kinds of magnetic devices, and is not limited to current measurement applications. 
   Most current monitoring systems for a-c (alternating-current) electric power systems utilize ordinary current transformers to provide input currents that are isolated from the electric power system conductors, similar to  FIG. 1 . A current-carrying conductor  4  is configured as a primary winding of a current transformer CT 1 , and is magnetically coupled to a magnetic core  1 . For clarification, the term “magnetic core” as used herein refers to a magnetic body having a defined relationship with one or more conductive windings. A secondary winding  2  is also magnetically coupled to magnetic core  1 . The phrase “magnetically coupled” is intended to mean that flux changes in a magnetic body are associated with an induced voltage in the winding, the induced voltage being proportional to the rate of change of magnetic flux that is coupled, in accordance with Faraday&#39;s Law. 
   A secondary electric current J 2  is induced in the secondary winding that is proportional to a primary electric current J 1 . The secondary current is isolated from the primary current and is smaller than the primary current by the turns ratio of the primary and secondary windings. The primary winding may consist of only one turn (as in  FIG. 1 ) or may have multiple turns wrapped around the magnetic core. The secondary winding usually consists of multiple turns wrapped around the magnetic core. 
   The accuracy of a current transformer is usually related to the coercive force of the magnetic core material (less is better), the cross sectional area of the magnetic core (bigger is better), the effective magnetic length of the magnetic core (shorter is better), any air gap in the magnetic core (less or none is better), and the “squareness” of the magnetic core material hysteresis curve (squarer may be preferred if not operating near saturation, otherwise characteristics that are not square may be preferred). Split-core current transformer cores generally have hysteresis curves that are less square than standard current transformer cores due to the small air gaps inherent in the design of split-core current transformers. 
   In order for the secondary current generated by a current transformer to be an accurate representation of the primary current, the impedance of the circuit connected to the secondary winding must be kept low so that current can flow freely. The impedance of the secondary circuit is often called the “burden.” The burden generally includes all impedances in the loop through which the secondary current flows, including stray winding impedances, stray impedances of connecting conductors, and the impedances of any devices connected in the loop (such as current-sensing resistors and relay operating coils). In order for a current transformer to drive a secondary current through a non-zero burden, a voltage must be induced in the secondary winding. The induced voltage is proportional to secondary current and is proportional to the burden, in accordance with Ohm&#39;s Law (the induced voltage equals the secondary current times the vector sum of all secondary loop impedances). The induced voltage is induced in the secondary winding by a fluctuating magnetic flux in the magnetic core (the instantaneous magnitude of induced voltage being proportional to the rate of change of magnetic flux, in accordance with Faraday&#39;s Law). The fluctuating magnetic flux is associated with an “exciting current” in accordance with well-known electromagnetic principles. The exciting current is often understood to have a magnetizing component and a core loss component. When utilized to measure alternating current with no d-c component, the exciting current accounts for the error in the secondary current, and may be referred to herein as an “exciting current error.” Generally speaking, the accuracy of a current transformer is inversely related to the burden of the secondary circuit. A higher burden causes the current transformer to operate with greater induced voltage, thereby increasing the exciting current error, thereby causing the secondary current to be less accurately proportional to the primary current. 
   With preferred current transformer operation, the amp-turns of the primary winding are largely canceled by the amp-turns of the secondary winding, so that the magnetomotive force acting on the current transformer core is relatively small. The net magnetomotive force acting on the core is equal to the difference in amp-turns of the primary winding and the secondary winding, and this difference is proportional to a secondary current error. 
   Speaking more precisely of current transformer operation, a secondary electric current error is proportional to the magnetomotive force acting on the magnetic core. The instantaneous value of the magnetomotive force is equal to the instantaneous difference between the primary electric current multiplied by the number of turns of the primary winding and the secondary electric current multiplied by the number of turns of the secondary winding. The secondary electric current error comprises a d-c component and an a-c component; the d-c component will be referred to as a d-c current error, and the a-c component will be referred to as an exciting current error. 
   Ordinary current transformers work properly only with alternating primary current. When a d-c component is present in the primary current, normal current transformer operation cannot maintain a d-c component in the secondary circuit, and a large d-c current error results. This d-c current error correlates to a large d-c magnetomotive force applied to the magnetic core, which causes the magnetic core to saturate, thereby adversely affecting current transformer operation. 
   A great many variations to the basic current transformer circuit have been developed in the prior art to improve current transformer accuracy for various applications. Some of these are summarized here:
         (a) Utilize an active load to sense current. An active load can have an effective burden of virtually zero Ohms, but this does not solve the problem of stray impedances contributing to the burden of the secondary circuit. The use of an active load to reduce current transformer burden is described in detail in U.S. Pat. No. Re. 28,851 to Milkovic (reissued 1976) for a “Current Transformer with Active Load Termination.”   (b)  FIG. 2  illustrates one form of a prior-art “zero-flux” current transformer. A sense winding  10  terminated in a high-impedance manner provides a voltage signal V 4  that is proportional to the rate of change of magnetic flux. By amplifying this signal and applying it in series with the secondary winding, the effective burden of the entire secondary circuit is reduced to near zero ohms. Magnetic flux oscillations in the current transformer core are reduced to near zero, and the exciting current required is reduced to near zero, thereby making secondary current more accurately proportional to primary current. The amplifier essentially provides the driving voltage necessary to drive loop current through secondary loop impedances so that the current transformer core does not need to generate this voltage via a changing flux. Higher gains in the amplifier circuit contribute to increased accuracy and smaller flux changes, though excessively high gain typically leads to instability and associated oscillations. This device provides very good accuracy for measurement of a-c current, but it cannot measure d-c current.   (c) In order to measure d-c current (or combined a-c and d-c current), Hall-effect current sensors are often used. These sensors typically insert a Hall-effect magnetic field sensor in a current transformer core air gap. In “open loop” devices, the magnetic field strength is used to estimate the primary current directly. “Closed loop” devices utilize a zero-flux concept similar to that described for  FIG. 2 . However, instead of using a sense winding (as in  FIG. 2 ), the Hall-effect element generates a voltage signal proportional to the magnetic field in the air gap. A high-gain amplifier circuit is used to drive secondary current to continuously nullify the magnetic field, which causes the secondary winding amp-turns to balance the primary winding amp-turns. This results in a secondary current that is proportional to the primary current. A current-sensing resistor in the secondary circuit normally provides a voltage signal that is proportional to secondary current. While these Hall-effect current sensors are widely used, their accuracy and stability over time are not adequate for many applications.   (d)  FIG. 3  shows another prior-art circuit that operates in a near-zero-flux manner. This type of “burden-reducing” circuit is described in U.S. patent application Ser. No. 09/713,921, filing date Nov. 15, 2000, by Edel. This patent application in its entirety is hereby incorporated by reference into this disclosure.    The circuit shown in  FIG. 3  uses the secondary current as an input to generate the compensation voltage required to drive secondary current. This circuit has the advantage of utilizing ordinary current transformers without the need for a sense winding or Hall-effect sensor. However, the circuit shown in  FIG. 3  can only be used to measure a-c current, and it is difficult to compensate for secondary loop impedance changes due to temperature changes. The associated patent describes how the control circuit can be modified to enable accurate measurement of d-c current, but the method used is dependent on brief periodic reset pulses applied to the magnetic core, during which time current cannot be measured.   (e) Many specialized current transformers with multiple windings and/or multiple cores have been developed. Many of these transformers have excellent accuracy. However, most of these specialized transformers are prohibitively expensive for many applications. Some devices having simple magnetic cores drive the core in and out of saturation to measure d-c current, often causing excessive noise on the primary circuit.       

   It is therefore an object of the present invention to provide an economical current sensor with the following properties:
         (a) Utilize an ordinary current transformer core.   (b) Provide for continuous measurement of a-c and d-c current.   (c) Have a high degree of stability over time and temperature.   (d) Have better accuracy than Hall-effect current sensors.   (e) Cause very little noise on the primary circuit.       

   Another object of the invention is to provide a way to measure magnetomotive force experienced by a magnetic body without utilizing a Hall-effect sensor. Other objects will become apparent from the description of the invention. 
   BRIEF SUMMARY OF THE INVENTION 
   An excitation means is connected to a winding that is magnetically coupled to a magnetic body. The magnetic body has nonlinear permeability and is influenced by a magnetomotive force. The excitation means generates an oscillating output so as to cause the magnetic flux within the magnetic body to oscillate at a predetermined frequency, preferably without causing the magnetic body to saturate. The oscillating flux is associated with an exciting current oscillating at the same frequency, and an excitation voltage oscillating at the same frequency (the excitation voltage is the voltage induced in the winding by the oscillating flux). The waveforms of the exciting current and excitation voltage are not the same due to the nonlinear permeability of the magnetic body. A difference of symmetry between these two excitation waveforms is dependent on the magnitude and polarity of the magnetomotive force. The invention utilizes known relationships between the difference of symmetry and the magnetomotive force to calculate the magnetomotive force. 
   For example, the excitation means may be a voltage source providing an oscillating output voltage having a symmetrical waveform, resulting in an oscillating exciting current that is only symmetrical when magnetomotive force has an average magnitude near zero. The dissymmetry of the exciting current is indicative of the polarity and average value of magnetomotive force experienced by the magnetic body. The dissymmetry may be used to measure the average magnetomotive force experienced by the magnetic body. Alternatively the dissymmetry may be used to generate an information signal for use as an input to a control system that controls the average magnetomotive force experienced by the magnetic body. 
   The excitation means may also be a current source providing an oscillating symmetrical output current, resulting in an oscillating excitation voltage. In this case, the dissymmetry of the excitation voltage is indicative of the polarity and average value of magnetomotive force experienced by the magnetic body. 
   Waveform dissymmetry is characterized by the Fourier series of the waveform having even harmonic components of the fundamental frequency. The preferred embodiment measures the second harmonic component, and calculates average magnetomotive force magnitude and polarity from the magnitude and phase angle of the second harmonic component. 
   For some applications, a difference of symmetry between excitation waveforms may also be used to measure the polarity and average magnitude of magnetic flux or a magnetic field. The correlation between waveform symmetry and a magnetic flux, magnetic field, and magnetomotive force is dependent on the configuration of the magnetic body and the magnetic material comprising the magnetic body. A simple proportional correlation usually exists between magnetic field strength and magnetomotive force for magnetic bodies with uniform magnetic characteristics. 
   Since magnetomotive force may be proportional to a single electric current flowing in a single winding, the invention may be used to measure an electric current in a non contact manner (without a proportional secondary current flowing in a secondary winding). 
   The invention may also be used to enable an ordinary current transformer to be used with a primary current having a d-c component. A voltage device is connected in series with the secondary circuit of the current transformer. The voltage device generates an output voltage in series with the secondary circuit. A control circuit controls the output voltage so as to cause the magnetic flux within the core of the current transformer to oscillate at a predetermined frequency. The flux oscillation preferably has low magnitude and low frequency, thereby minimizing noise on the primary circuit. The flux oscillation is associated with a small exciting current of the same frequency. The dissymmetry of this exciting current is indicative of the polarity and average value of magnetomotive force experienced by the current transformer core. A nonzero average magnetomotive force is due to a d-c current error (the difference in the d-c amp-turns of the primary winding and the d-c amp-turns of the secondary winding). The exciting current dissymmetry is detected and used to adjust the output voltage of the voltage device so as to maintain the average value of magnetomotive force near zero. This minimizes the d-c current error, thereby enabling the current transformer to operate with d-c primary current. 
   For current transformer operation with mixed a-c and d-c current, the preferred embodiment controls the output voltage to minimize flux changes normally associated with alternating-current operation. Otherwise, the preferred flux oscillations at the predetermined frequency would be corrupted, thereby corrupting the exciting current signal and adversely affecting operation. The preferred embodiment for current measurement ( FIG. 10  &amp;  FIG. 11 ) combines two prior-art compensation methods with sensing of average magnetomotive force. This embodiment automatically compensates for secondary loop impedance changes due to temperature and time, and has very good accuracy and stability. 
   Usually, it is an “average value” of magnetomotive force that is determined by the invention, since the flux oscillations required by the invention (at the predetermined frequency) are associated with an oscillation of magnetomotive force (also at the predetermined frequency, and proportional to the exciting current). These oscillations at the predetermined frequency are not usually included in the measurement of magnetomotive force. However, it is possible to add the magnetomotive force associated with the exciting current to the measurement and thereby obtain an information signal that instantaneously correlates to the instantaneous magnitude of magnetomotive force. The preferred embodiment, however, does not do this. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  illustrates a simple prior-art current transformer circuit that utilizes a current-sensing resistor to produce a voltage that is approximately proportional to primary current. 
       FIG. 2  illustrates a prior-art “zero-flux” current transformer circuit that utilizes a voltage device in the secondary circuit to improve accuracy for a-c current measurement applications. A sense winding  10  develops a voltage V 4  that is proportional to the rate of change of magnetic flux in the magnetic core. This voltage is amplified and added in series with the secondary circuit so as to sharply reduce flux changes. 
       FIG. 3  illustrates another prior-art circuit that utilizes a voltage device in the secondary circuit to improve accuracy for a-c measurement applications. This circuit utilizes a signal that is proportional to secondary current to control the voltage device with a proportional plus derivative type of control. When properly adjusted, the voltage device provides most of the voltage necessary to drive secondary current through secondary loop impedances, thereby sharply reducing flux changes in the current transformer core. 
       FIG. 4A  shows a simple circuit that will be used to explain the fundamental principles of the invention. 
       FIG. 4B  through  FIG. 6C  show hysteresis curves and waveforms that illustrate how the invention utilizes nonlinearities of a magnetic core to measure magnetomotive force. 
       FIG. 7  shows a block diagram of a first embodiment of the invention. 
       FIG. 8  is a detailed schematic diagram of the control circuit of  FIG. 7 . 
       FIG. 9A  through  FIG. 9L  show waveforms associated with the control circuit of  FIG. 7  and  FIG. 8  (these waveforms are also applicable to some subsequent figures). 
       FIG. 9M  shows a waveform associated with the control circuit of  FIG. 10  and  FIG. 11 . 
     FIG.  FIG. 9N  shows a waveform associated with the circuit of  FIG. 12 . 
       FIG. 10  is a block diagram of a second embodiment of the invention. This embodiment is similar to  FIG. 7 , but additionally includes two kinds of prior-art flux control methods to improve operation. 
       FIG. 11  shows a detailed schematic diagram of the control circuit of  FIG. 10 . 
       FIG. 12  illustrates how the exciting current error may be approximated and subtracted from the secondary current signal to provide a signal that is more accurately proportional to the primary current. 
       FIG. 13  shows an alternate embodiment utilizing a digital signal processor rather than analog electronics. 
       FIG. 14  shows how an inexpensive microcontroller may be combined with analog controls in another alternate embodiment of the invention. 
       FIG. 15  is a simplified embodiment of the invention that operates in an open-loop configuration (no feedback) for measurement of slowly-changing magnetomotive force (or d-c primary current). The output of lowpass filter  17  is utilized directly as a measurement of magnetomotive force (or d-c primary current). This is an open-loop version of the invention that requires less operating power, but lacks the accuracy and range of the closed-loop versions. 
       FIG. 16  shows an approximate response curve for the configuration shown in  FIG. 15 . 
       FIG. 17  shows an alternate embodiment intended to generalize some aspects of the invention.  FIG. 17  illustrates how current sources may be used in place of voltage sources, how an exciting means may utilize a dedicated winding separate from a control winding, how the invention may be applied to magnetic bodies other than magnetic cores of current transformers, and how a general-purpose Proportional plus Integral plus Derivative (PID) controller may be utilized with the invention. 
   

   DETAILED DESCRIPTION OF THE DRAWINGS 
     FIG. 1  illustrates a simple prior-art current measurement circuit. A current transformer CT 1  comprises a magnetic core  1  and a secondary winding  2  magnetically coupled to the magnetic core. An electric power system conductor  4  with an insulating covering  3  is configured as a primary winding with only one turn, with a primary electric current J 1  flowing. Though shown with one end disconnected, power system conductor  4  is normally connected as part of an electric power system. Though a toroid is shown for magnetic core  1 , current transformers have many different kinds of magnetic cores. The number of winding turns shown for secondary winding  2  is for illustration only, and may vary widely depending on the particular application. Current transformers are also available with wound primary windings or with bar-type primary configurations. 
   A secondary electric current J 2  is caused to flow by the transformer action of current transformer CT 1  (in accordance with well-known current transformer principles). If current transformer CT 1  were to have ideal properties, then secondary current J 2  would be exactly (and instantaneously with no phase angle error) proportional to primary current J 1 , with magnitude reduced exactly by the turns ratio of the current transformer. However, since current transformers generally do not have ideal properties, there is an exciting current error in secondary current J 2 , and secondary current J 2  is only approximately proportional to primary current J 1 . 
   In  FIG. 1 , a current-sensing resistor R 1  with small resistance is connected in series with secondary winding  2  so that current J 2  flows through resistor R 1 . A voltage signal V 1  is the voltage across resistor R 1 , and is proportional to current J 2 . Since secondary current J 2  is approximately proportional to primary current J 1 , voltage signal V 1  is approximately proportional to primary current J 1 . Voltage signal V 1  is often used as an input to a current monitoring system, energy meter, protective relay, or similar device. 
   Voltage V 2  is the secondary winding voltage that is measurable at the terminals of secondary winding  2 . Not taking into account stray impedances, in  FIG. 1  voltage V 2  is equal to voltage V 1  in magnitude, and is opposite in polarity. Voltage V 2  is generated by the current transformer as a result of changing magnetic flux associated with exciting current. 
     FIG. 2  illustrates a prior-art “zero-flux” current transformer circuit that utilizes a controllable voltage device  5 A in the secondary circuit to improve accuracy for a-c current measurement applications. Though an operational amplifier circuit is shown, voltage device  5 A may be any kind of device that can produce a controllable output voltage. A current transformer CT 2  is similar to current transformer CT 1  in  FIG. 1 , except that now a sense winding  10  is added to magnetic core  1 . Sense winding  10  is terminated in a high-impedance manner and therefore has almost zero current flowing in it. With almost no current flowing, a voltage signal V 4  that is generated across sense winding  10  is proportional to the rate of change of magnetic flux in the magnetic core (in accordance with Faraday&#39;s Law). Voltage signal V 4  is connected to an operational amplifier  6  by a conductor  7 . Resistors R 2 A and R 3 A are connected so as to control the gain of operational amplifier  6 . The resistance of resistor R 2 A is selected to be much greater than resistor R 3 A so that the voltage gain of the amplifier is relatively large. Voltage V 3  is the output voltage of the voltage device, and is simply voltage signal V 4  amplified. Voltage V 3  is connected in series with the secondary circuit, and provides most of the driving voltage required to drive current J 2  through the secondary loop impedances. These loop impedances include resistor R 1 , stray winding resistance, stray winding reactance, and stray impedances of connecting wires. This sharply reduces flux changes in magnetic core  1 , thereby sharply reducing the exciting current error, thereby making secondary current J 2  more accurately proportional to primary current J 1 . 
     FIG. 3  shows another prior-art circuit that operates in a near-zero-flux manner. This “burden-reducing” circuit uses the secondary current as an input to generate the compensation voltage required to drive secondary current. This circuit has the advantage of utilizing ordinary current transformers without the need for a sense winding or Hall-effect sensor. 
   Current transformer CT 1  and current-sensing resistor R 1  are the same as shown in  FIG. 1 . A controllable voltage device  5 B produces an output voltage V 3  similar to voltage device  5 A of  FIG. 2 . In  FIG. 3 , voltage V 3  is controlled based on voltage signal V 1 . Resistor R 2 B, Resistor R 3 B, and Capacitor C 1  are configured to cause voltage V 3  to have a component that is proportional to secondary current (which is proportional to voltage signal V 1 ) and to have a component that is proportional to the rate-of-change of secondary current. The component that is proportional to secondary current provides the driving voltage required to drive secondary current through secondary resistances (such as resistor R 1 , winding resistance, and stray resistances of other conductors). The component that is proportional to the rate-of-change of secondary current provides the driving voltage required to drive secondary current through secondary reactive impedances (such as stray inductances in the secondary winding and other conductors). Resistors R 2 B and R 3 B are shown as variable to facilitate calibration for optimum operation. Adjusting resistor R 3 B calibrates for the ratio of secondary loop resistance to inductance, while adjusting resistor R 2 B adjusts the gain of the amplifier. For secondary circuits in which stray inductances are small, capacitor C 1  may be omitted, in which case the gain of the amplifier should simply be set to a value slightly less than the total loop resistance divided by the resistance of R 1 . 
     FIG. 4A  shows a simple magnetic circuit that will be used to explain how the invention utilizes nonlinearities of a magnetic body to measure the average value of a magnetomotive force. A magnetic body  1 A having nonlinear permeability is magnetically coupled to a primary winding  4 A and a secondary winding  2 A. A d-c current J 1 A flowing in primary winding  4 A produces a magnetomotive force in magnetic body  1 A that is to be measured. 
   An oscillating excitation means  5 C is connected to secondary winding  2 A. Excitation means  5 C may be configured as a voltage source (which produces a defined output voltage and allows output current to be dependent on the load) or a current source (which produces a defined output current, and allows the output voltage to be dependent on the load), or may be any kind of electric energy source capable of producing an oscillating output at a predetermined frequency. 
   For the present explanation, source  5 C will be considered to be a voltage source that produces a sinusoidal output voltage V which oscillates at a predetermined frequency. This causes an exciting current J 2 A to flow, which also oscillates at the predetermined frequency. For simplicity of explanation, the effect of stray impedances will be considered negligible in the present discussion, so that voltage V is equal to the induced excitation voltage in winding  2 A. The oscillating excitation voltage is associated with an oscillating magnetic flux Φ in magnetic core  1 A. More specifically, the rate of change of flux Φ is proportional to voltage V (in accordance with Faraday&#39;s Law). 
     FIG. 4B  shows a hysteresis curve for magnetic body  1 A, applicable for the case of d-c current J 1 A initially being zero amps. The horizontal axis represents magnetomotive force F, and varies from negative values on the left to positive values on the right with a linear scale. The vertical axis represents magnetic flux Φ, and varies from negative values on the lower side to positive values above with a linear scale. 
   The hysteresis cycle associated with voltage V is shown bold, while a saturating hysteresis cycle is shown lighter. Voltage V causes flux to oscillate by an amount ΔΦ. This correlates with an oscillation of the magnetomotive force by an amount of ΔF. 
   It should be noted that the excitation means preferably does not drive the magnetic body into saturation, so the magnitude of flux oscillations is preferably much smaller than saturation levels. The predetermined frequency may be any frequency that is suitable for a particular application and a particular magnetic body. 
     FIG. 4C  through  FIG. 4E  show sample waveforms correlating to the hysteresis curve of  FIG. 4B . A time reference T 1  is included to facilitate comparison of waveform relationships. For each waveform, time t increases from left to right. 
   In  FIG. 4C , Voltage V is shown as sinusoidal, which causes the waveform of flux Φ to be sinusoidal and lagging voltage V. The waveform of magnetomotive force F is shown in  FIG. 4D , and, in this case, the waveform of exciting current J 2 A is the same as the waveform shown for magnetomotive force F (since current J 1 A is zero). Since magnetic body  1 A has nonlinear characteristics, magnetomotive force F is seen to be distorted (not strictly sinusoidal). However, since the hysteresis characteristics of magnetic body  1 A are symmetrical about the origin, the waveform of magnetomotive force F has half-wave symmetry. 
   Waveforms having half-wave symmetry satisfy the equation:
 
 f ( t )=− f ( t±T/ 2)
 
where f(t) represents the repeating waveform as a function of time, and T is the time period of one cycle of the wave. With reference to well-known Fourier analysis techniques, it is known that waveforms having half-wave symmetry contain no even harmonic components of the fundamental frequency.  FIG. 4E  illustrates this, by showing the second harmonic F 2  of magnetomotive force F to have a continuous magnitude of zero.
 
     FIG. 5A  shows another hysteresis curve for magnetic body  1 A, applicable for the case of d-c current J 1 A having a positive magnitude. The hysteresis cycle associated with voltage V is shown bold, while a saturating hysteresis cycle is shown lighter. Voltage V still causes flux to oscillate by an amount ΔΦ. (The waveforms associated with flux Φ and voltage V are the same as shown in  FIG. 4C .) The hysteresis loop is now offset from the origin due to the nonzero magnetomotive force associated with, current J 1 A. The hysteresis loop is now seen to be somewhat unsymmetrical, due to the nonlinear permeability of magnetic core  1 A. 
     FIG. 5B  shows the waveform of magnetomotive force F as the sum of magnetomotive forces associated with current J 1 A and exciting current J 2 A. Exciting current J 2 A no longer has half-wave symmetry, and therefore now contains even harmonic components of the predetermined frequency.  FIG. 5C  shows the waveform of the second harmonic component F 2 . 
     FIG. 6A  shows another hysteresis curve for magnetic body  1 A, applicable for the case of d-c current J 1 A having a negative magnitude. The hysteresis cycle associated with voltage V is shown bold, while a saturating hysteresis cycle is shown lighter. Voltage V still causes flux to oscillate by an amount ΔΦ. (The waveforms associated with flux Φ and voltage V are the same as shown in  FIG. 4C .) The hysteresis loop is again offset from the origin due to the nonzero magnetomotive force associated with current J 1 A, but now in the opposite direction. The hysteresis loop is again seen to be unsymmetrical, due to the nonlinear permeability of magnetic core  1 A. 
     FIG. 6B  shows the waveform of magnetomotive force F as the sum of magnetomotive forces associated with current J 1 A and exciting current J 2 A. Exciting current J 2 A again does not have half-wave symmetry, and therefore again contains even harmonic components of the predetermined frequency.  FIG. 6C  shows the waveform of the second harmonic component F 2 . The waveform in  FIG. 6C  is similar to the waveform in  FIG. 5C , except that the phase angle is changed by 180 degrees (or the second harmonic could be considered to be inverted or have opposite polarity). This difference in phase angle of the second harmonic component is a known correlation used by the preferred embodiment to determine the polarity of the magnetomotive force. The magnitude of the second harmonic component has a known correlation to the magnitude of the magnetomotive force (as later discussed for  FIG. 17 ), and is used by the preferred embodiment to determine the magnitude of the magnetomotive force. 
   The symmetry relationships illustrated in  FIGS. 4C ,  4 D,  5 B, and  6 B are one example of how the invention may utilize predetermined symmetry relationships to measure magnetomotive force. 
   While voltage V has been shown as sinusoidal, the waveform is not critical. A sinusoidal waveform has been chosen to minimize noise. A symmetrical waveform is preferred so that a difference of symmetry between the excitation voltage waveform and the exciting current waveform is easily detectable. 
   The invention can also utilize a current source for exciting means  5 C in  FIG. 4A . With a current-source configuration, the current source preferably drives a symmetrical oscillating exciting current, and an unsymmetrical excitation voltage results. The dissymmetry of the excitation voltage is sensed in a manner that is similar to sensing the dissymmetry of unsymmetrical exciting current (as described for  FIGS. 5A through 6C ). 
   In general, excitation means  5 C can be any kind of electric energy source capable of producing an oscillating output at a predetermined frequency. In the general case of an output signal that may not have a symmetrical waveform, then the exciting current is characterized by a first waveform, and the excitation voltage is characterized by a second waveform, and a difference of symmetry between the two waveforms is dependent on the magnitude and polarity of the magnetomotive force and the nonlinear permeability of the magnetic body. By utilizing predetermined symmetry relationships, the magnetomotive force may be determined from characteristics of one or both waveforms. 
     FIG. 7  illustrates an embodiment of the invention that enables an ordinary current transformer to provide a secondary current that is proportional to a primary d-c current. By controlling magnetomotive force, buildup of magnetic flux is prevented and an ordinary current transformer can operate with a d-c primary current. 
   While embodiment shown in  FIG. 7  is intended to measure d-c current, it may be used to measure currents having alternating current components with frequencies slower than the predetermined frequency for excitation (which is 40 Hz. in  FIG. 15 ). The practical frequency limit is dependent on properties of magnetic core  1 , response characteristics of control circuit  11 D, and, of course, the excitation frequency utilized. Using a faster excitation frequency is a simple way to extend the usable frequency range. 
   Current transformer CT 1  is the same as shown in  FIG. 1  and  FIG. 3 . Though a toroid is shown for magnetic core  1 , current transformers have many different kinds of magnetic cores (the invention is applicable to virtually any kind of current transformer with any form of magnetic core, including split-core transformers if the effects of air gaps are minimized). The number of winding turns shown for secondary winding  2  is for illustration only, and may vary widely depending on the particular application. Current transformers are also available with wound primary windings or with bar-type primary configurations. 
   Similar to  FIG. 3 , a controllable voltage device  5  is connected in series with secondary winding  2 . Voltage device  5  can be any kind of electrical energy source capable of producing a controllable output voltage in a current transformer secondary circuit. A simple embodiment of voltage device  5  is shown, comprising operational amplifier  6  and gain-control resistors R 2  and R 3 . Similar to  FIGS. 1 ,  2 , and  3 , secondary current J 2  flows through current-sensing resistor R 1 , thereby producing voltage signal V 1 . Resistor R 1  is a sensing means for sensing secondary electric current J 2 . 
   A control circuit  11  continuously causes voltage device  5  to produce an oscillating excitation voltage, and continuously senses dissymmetry of exciting current, and continuously adjusts output voltage V 3  so as to minimize d-c current error, thereby causing secondary current J 2  to be continuously proportional to primary current J 1 . Control circuit  11  may be referred to as a calculating means or a control means. 
   In the preferred embodiment, operational amplifier  6  is a power op amp, Burr-Brown model OPA548T. This device has an adjustable current-limit feature and is rated for up to three amps of continuous current (other models are available with higher ratings). A “snubber circuit” (not shown) should be connected to the output to improve stability (as recommended by the device&#39;s data sheet). A “snubber circuit” that has been successfully utilized is a ten ohm resistor in series with a 0.1 microfarad capacitor, connected between the operational amplifier output and the grounded conductor. An additional resistor (not shown) is necessary to set the current limit of the device, in accordance with the device&#39;s data sheet. 
   Voltage device  5  primarily needs to provide current gain, and a small voltage gain works well. The values chosen for resistors R 2  and R 3  are not critical. In the preferred embodiment resistor R 3  is 2.0 kilohm, and resistor R 2  is 200 ohms (causing voltage device  5  to have a voltage gain of about 1.1). 
   In the preferred embodiment, magnetic core  1  is a tape-wound core made with 7-mil grain-oriented silicon steel processed for minimum coercive force. The core has an internal diameter of 0.5 inch, an outside diameter of 1.5 inches, and a thickness of 0.5. Winding  2  is 110 turns of 26 AWG copper magnet wire, and has a winding resistance of about 0.9 ohms. 
   Current-sensing resistor R 1  is rated 1.0 ohm in the preferred embodiment. 
   Control circuit  11  may be implemented with analog electronics or digital electronics, or a combination of the two. The function blocks shown in  FIG. 7  are implemented with analog electronics in  FIG. 8 , but will first be discussed from a functional point of view. 
   An oscillator circuit comprises a square-wave oscillator  19 , a frequency divider  20 , and a bandpass filter  21 . This oscillator circuit, together with voltage device  5 , is an excitation means for causing magnetic flux in the magnetic body to oscillate at the predetermined frequency. 40 Hz. is a predetermined operating frequency for this control circuit. The frequency utilized is not critical, and has been chosen to minimize noise associated with the circuit and to not conflict with electric power system frequencies (50 or 60 Hz. and related harmonics). 
   Oscillator  19  produces a square-wave output having an 80 Hz. frequency, which is twice the predetermined frequency. This output is used as a control signal  9 D to a biphase amplifier  16 , and is also an input signal to frequency divider  20 . 
   Frequency divider  20  provides a 40 Hz. square-wave signal  9 E that is precisely half the frequency of signal  9 D. 
   Bandpass filter  21  has a narrow bandpass characteristic set close to 40 Hz. This filter changes signal  9 E from a square waveform to a sinusoidal waveform signal  9 F, and adjusts the phase angle so that signal  9 D (the biphase amplifier control signal) is timed properly relative to signal  9 F. The magnitude of signal  9 F is modified as it is passed through signal adder  14 , and becomes one of several signal components in signal  9 L and voltage V 3 . Signal  9 F is a symmetrical excitation signal that causes an oscillation of magnetic flux in magnetic core  1 , resulting in an exciting current component in secondary current J 2 . 
   In the preferred embodiment, the excitation voltage component of output voltage V 3  does not drive magnetic body  1  into saturation. The preferred embodiment utilizes nonlinearities in core permeability below saturation levels. The magnitude of excitation voltage required for good operation is dependent on core permeability characteristics, the core cross-sectional area, the number of winding turns, the excitation frequency, and the sensitivity of the second-harmonic detection circuits. Relatively small excitation voltage and low frequency are usually preferred to minimize noise coupled to the primary circuit. While the preferred embodiment utilizes an excitation frequency of only 40 Hz., some applications will benefit from utilizing much higher frequencies, such as 1000 Hz., 10,000 Hz., or even higher. 
   Secondary current J 2  flowing through resistor R 1  causes voltage V 1  to be proportional to secondary current J 2 . Signal  9 B is voltage V 1 , and is buffered by buffer  12 , so that signal  9 C is also equal to voltage V 1 . Buffer  12  may be deleted if bandpass filter  15  has sufficiently high input impedance so that voltage V 1  is not adversely affected. 
   80 Hz. bandpass filter  15  amplifies the second harmonic of exciting current (relative to other frequency components) and produces signal  9 G for input to biphase amplifier  16 . It may also be said that bandpass filter  15  is used for “filtering” the second harmonic component. 
   Biphase amplifier  16  has a gain of plus or minus one, depending on the polarity of control signal  9 D received from the oscillator circuit. Biphase amplifier  16  is used for detecting the magnitude and phase angle of the second harmonic component. Operation of biphase amplifier  16  is timed (by shifting the phase angle of signal  9 F) so that the polarity of the gain is switched approximately at every zero-crossing point of the second harmonic component of exciting current. In this configuration, biphase amplifier  16  operates like a synchronous rectifier for the second harmonic component of exciting current. This causes signal  9 H to have an average magnitude approximately proportional to the magnitude of the exciting current second harmonic component, with polarity corresponding to the phase angle of the second harmonic component. 
   Lowpass filter  17  smooths signal  9 H so that signal  9 J is approximately the average value of signal  9 H. 
   Signal  9 J has a strong correlation to the average magnetomotive force experienced by the magnetic core. For small values of magnetomotive force (for many types of magnetic bodies), signal  9 J is approximately proportional to magnetomotive force. Signal  9 J is a bipolar signal, with its polarity corresponding to the polarity of magnetomotive force. This known correlation may be further quantified for a particular configuration by performing simple tests (with the feedback loop open), and developing a graph similar to  FIG. 16 . 
   In  FIG. 7 , signal  9 J is used as a feedback signal for a proportional plus integral type of control to minimize the average magnetomotive force. Signal  9 J is fed back to the secondary circuit through signal adder  14  for proportional control. Signal integrator  18  integrates signal  9 J, producing signal  9 K, which is fed back via signal adder  14 , thereby providing integral control. Signal integrator  18  (and signal  9   k ) may be omitted for some applications. It is included in the preferred embodiment to minimize error associated with the control loop. 
   Signal adder  14  may have different gains for each input signal. The instantaneous magnitude of signal  9 L is therefore a weighted sum of the instantaneous magnitudes of signals  9 F,  9 J, and  9 K. Signal adder  14 , signal integrator  18 , and voltage device  5  may be described as a control system that receives signal  9 J as an information signal and that controls the magnetomotive force experienced by magnetic core  1 . 
     FIG. 8  is a detailed schematic diagram showing a preferred embodiment of control circuit  11  in  FIG. 7 . The embodiment shown in  FIG. 8  utilizes analog electronic calculating means.  FIG. 8  shows just one of many possible embodiments of control circuit  11 . Resistors critical to filter or timing functions should have 1% tolerance or better and have good temperature stability. Capacitors in filter or timing circuits should have 5% tolerance or better and have good temperature stability. All operational amplifiers utilized in control circuit  11  are Texas Instruments type TLO64ACN with JFET inputs. 
   A d-c power supply (not shown) provides operating power to terminals  11 A and  11 B, with positive voltage set to +6 volts and negative voltage set to −6 volts. Power supply connections to the operational amplifiers and analog switch  16 F are not shown. Capacitors  11 C,  11 D,  11 E, and  11 F are included to reduce power supply noise. Capacitors  11 C and  11 D are rated 0.1 microfarad, and capacitors  11 E, and  11 F are electrolytic capacitors rated 10 microfarads. 
   Buffer  12  is simply an op amp  12 A configured as a voltage follower, so that the output voltage is continuously the same as the input voltage. 
   Signal adder  14  is built around an op amp  14 A. The output is an inverted weighted sum of the inputs. Component values are:
         Resistor  14 B: 27 kilohm   Resistor  14 C: 470 kilohm   Resistor  14 D: 1.0 megohm   Resistor  14 E: 24 kilohm       

   80 Hz. bandpass filter  15  is built around an op amp  15 A. The configuration shown inverts the input signal (the 80 Hz. component). Component values are:
         Resistor  15 B: 750 kilohm   Capacitor  15 C: 0.10 microfarad   Resistor  15 D: 10.0 kilohm   Capacitor  15 E: 0.10 microfarad   Resistor  15 F: variable to 1.0 kilohm (set for 80 Hz. bandpass operation with zero degrees phase shift at 80 Hz., approx. 720 ohms)       

   Biphase amplifier  16  is built around an op amp  16 A. Component values are:
         Resistor  16 B: 20.0 kilohm   Resistor  16 C: 20.0 kilohm   Resistor  16 D: 20.0 kilohm   Resistor  16 E: 20.0 kilohm   Switch  16 F: An analog switch, Fairchild model MM74HC4066N (only one of four switches included is used).       

   When control signal  9 D is high, switch  16 F is closed, and biphase amplifier  16  inverts signal  9 G. When control signal  9 D is low, switch  16 F is open and signal  9 H is approximately equal to signal  9 G (not inverted). 
   Lowpass filter  17  is built around an op amp  17 A. The configuration shown inverts the input signal. Component values are:
         Capacitor  17 B: 0.10 microfarad   Resistor  17 C: 1.00 megohm   Resistor  17 D: 499 kilohm   Resistor  17 E: 1.00 megohm   Capacitor  17 F: 0.10 microfarad   Resistor  17 G: 51 kilohm   Potentiometer  17 H: 100 kilohm   Resistor  17 J: 100 ohm       

   Resistors  17 G and  17 J and potentiometer  17 H are included for d-c offset correction of the operational amplifiers. Potentiometer  17 H should be set so that secondary current J 2  has an average value of zero when primary current J 1  is zero. 
   Signal integrator  18  is built around an op amp  18 A. This is a noninverting integrator that includes a proportional component in the output, as well as the integral of the input signal. Component values are:
         Capacitor  18 B: 1.0 microfarad   Resistor  18 C: 240 kilohm       

   80 Hz. square-wave oscillator  19  is built around an op amp  19 A. Component values are:
         Resistor  19 B: variable to 1.0 megohm (set for 80 Hz. output)   Resistor  19 C: 49.9 kilohm   Capacitor  19 D: 0.10 microfarad   Resistor  19 E: 49.9 kilohm   Resistor  19 F: 29.4 kilohm       

   Frequency divider  20  is built around a type J-K set-reset flip-flop  20 A (Fairchild model CD4027BCN, only one of two flip-flops is used). Component values are:
         Resistor  20 B: 10 kilohm   Resistor  20 C: 1.0 kilohm       

   Resistors  20 B and  20 C function as a voltage divider. 
   40 Hz. bandpass filter  21  is built around an op amp  21 A. The configuration shown inverts the input signal. Component values are:
         Resistor  21 B: 1.00 megohm   Capacitor  21 C: 0.10 microfarad   Resistor  21 D: 75.0 kilohm   Capacitor  21 E: 0.10 microfarad   Resistor  21 F: variable to 10 kilohm (set for approximately 40 Hz. bandpass operation and phase shift as described for  FIG. 9F , approx. 1550 ohms)       

     FIG. 9A  through  FIG. 9L  show waveforms associated with the control circuit of  FIG. 7  and  FIG. 8  (these waveforms are also applicable to some subsequent figures). Some of the waveforms shown are most easily observed in a test circuit by opening the feedback loop by disconnecting signals  9 J and  9 K from signal adder  14 . Time references T 2 , T 3 , and T 4  are included to facilitate comparison of the various waveforms. Time t increases from left to right for each waveform. 
     FIG. 9A  shows primary current J 1  as a d-c current. 
     FIG. 9B  shows signal  9 B, which is the same as voltage V 1 , which is proportional to secondary current J 2 . This is seen to be proportional to primary current J 1  with an exciting current error added. 
     FIG. 9C  shows signal  9 C, which is the same as signal  9 B. 
     FIG. 9D  shows signal  9 D, an 80 Hz. square wave. 
     FIG. 9E  shows signal  9 E, a 40 Hz. square wave synchronized with signal  9 D. 
     FIG. 9F  shows signal  9 F, a 40 Hz. sinusoid which is inverted and phase-shifted somewhat relative to signal  9 E. The amount of phase shift shown is approximately an amount required to align the zero crossing points of signal  9 D with the zero crossing points of the second harmonic in signals  9 B,  9 C and  9 G. Signal  9 F becomes the excitation voltage component of output voltage V 3  (after its amplitude is modified by signal adder  14  and voltage device  5 ). Its magnitude as a component of voltage V 3  is preferably about 0.5 volts peak-to-peak. This causes a 40 Hz. sinusoidal noise signal on primary conductor  4  of less than 5 millivolts peak-to-peak. 
     FIG. 9G  shows signal  9 G, which is the output of the 80 Hz. bandpass filter. The waveform of this signal will show considerable variation depending on the magnitude and waveform of primary current J 1 . The signal shown has the second harmonic of exciting current amplified relative to other components of signal  9 C. 
     FIG. 9H  shows signal  9 H, which is the output of biphase amplifier  16 . Whenever signal  9 D is high, signal  9 G is inverted by the biphase amplifier. Signal  9 H has an average value that depends on the magnitude and polarity of the second harmonic of the exciting current. 
     FIG. 9J  shows signal  9 J, which is approximately the average value of signal  9 H. 
     FIG. 9K  shows signal  9 K, which is the integral of signal  9 J. 
     FIG. 9L  shows signal  9 L, which is the weighted (and inverted) sum of signals  9 F,  9 J, and  9 K. In the case shown, signal  9 F dominates, and signal  9 L looks much like signal  9 F inverted. 
     FIG. 9M  will be discussed below along with  FIG. 10  and  FIG. 11 . 
     FIG. 9N  will be discussed below along with  FIG. 12 . 
     FIG. 10  is a variation of  FIG. 7 , in which two types of prior-art compensation methods are utilized: a “zero-flux” method and a “burden-reducing” method (discussed previously under “Background of the Invention” and for  FIG. 2  and  FIG. 3 ). Variations of the embodiment shown in  FIG. 10  may include just one of these compensation methods, rather than both. By controlling magnetomotive force and reducing normal (but undesirable) flux oscillations, buildup of magnetic flux is prevented and an ordinary current transformer can operate with a-c and d-c primary currents. 
   The embodiment shown in  FIG. 10  is well-suited to measuring a primary current having any combination of a-c and d-c components, including a primary current having one or more frequency components faster than the predetermined frequency (which is 40 Hz. in  FIG. 10 ). 
   Most function blocks in control circuit  11 A (in  FIG. 10 ) are the same as in control circuit  11  in  FIG. 8 , and function the same as previously described. New function blocks include feedback control  13 , lowpass amplifier  22 , and signal subtracter  23 . 
   The “burden-reducing” method is implemented by adding feedback control  13 . Signal  9 M has a component that is proportional to secondary current J 2 , and may optionally also have a component that is proportional to the rate of change of secondary current J 2 . The waveforms shown in  FIG. 9A  through  FIG. 9L  are applicable to  FIG. 10 . Additionally,  FIG. 9M  shows a waveform for signal  9 M, which is approximately proportional to signal  9 C, but inverted (the rate of change component is too small to be observed). 
   A “zero-flux” compensation method is implemented by adding sense winding  10  which develops voltage V 4  which is proportional to the rate of change of flux in magnetic core  1 . The preferred embodiment uses about 55 turns of 26 AWG copper magnet wire for winding  10 . Current transformer CT 2  is similar to current transformer CT 1  except for this added sense winding. Signal  9 P is the same as voltage V 4 . 
   A lowpass amplifier  22  receives and amplifies signal  9 P and filters out high-frequency components, so that signal  9 R is an amplified version of signal  9 P without high-frequency components that may cause instability in the loop. Lowpass amplifier  22  has relatively high input impedance so that voltage V 4  is not adversely affected. 
   A signal subtracter  23  subtracts signal  9 F (the 40 Hz. excitation voltage required by the invention) from signal  9 R, effectively subtracting the preferred 40 Hz. voltage induced in winding  10 . Signal  9 S is then proportional to the rate of change of unwanted flux changes. Signal  9 S is amplified by signal adder  14  and fed back to the secondary circuit as part of signal  9 L and voltage V 3 . In this way, unwanted flux changes due to changes in primary current J 1  are further reduced, while allowing the 40-Hz. preferred flux oscillations to be unaffected. 
   By reducing undesirable changes in magnetic flux associated with normal current transformer operation, the configuration of  FIG. 10  can operate with any combination of a-c and d-c currents. 
     FIG. 11  shows a schematic diagram for control circuit  11 A. Most parts are the same as  FIG. 8 , and function the same as previously described. New parts include feedback control  13 , lowpass amplifier  22 , and signal subtracter  23 . Preferred embodiments are as follows: 
   All operational amplifiers shown in  FIG. 11  are Texas Instruments type TLO64ACN with JFET inputs. 
   Feedback control  13  is built around an op amp  13 A. Component values are:
         Resistor  13 B: 10 kilohm   Capacitor  13 C: 1200 picofarad   Resistor  13 D: 15 kilohm   Capacitor  11 G: 100 picofarad       

   Resistor  13 B determines the amount of proportional feedback, and capacitor  13 C determines the amount of derivative (rate-of-change) feedback. The ratio of capacitance to resistance of these two devices should correlate to the ratio of inductance to resistance of secondary loop impedances. Since secondary loop inductance is small compared to secondary loop resistance, derivative feedback is not critical to good operation (in the preferred embodiment), and capacitor  13 C may be omitted. The 1200 picofarad rating of capacitor  13 C is a very rough approximation of the amount of derivative feedback required for optimum operation. 
   Lowpass amplifier  22  is built around an op amp  22 A. Component values are:
         Resistor  22 B: 2.00 kilohm   Capacitor  22 C: 0.01 microfarad   Resistor  22 D: 32.4 kilohm   Resistor  22 E: 16.2 kilohm       

   Signal subtracter  23  is built around an op amp  23 A. Component values are:
         Resistor  23 B: 20.0 kilohm   Resistor  23 C: 20.0 kilohm   Potentiometer  23 D: 50 kilohm       

   Potentiometer  23 D is configured as an adjustable voltage divider. It should be adjusted so that the 40 Hz. excitation signal (which is proportional to signal  9 F) is subtracted accurately from signal  9 R, resulting in signal  9 S having little or no 40 Hz. component). 
   Resistors  14 F and  14 G are also new. Resistor  14 G is rated 4.64 kilohm. Resistor  14 F is variable to 50 kilohm. Variable resistor  14 F should be set for best burden-reducing operation. With stray secondary loop resistances in the preferred embodiment approximately equal to 0.9 ohms (mostly secondary winding resistance), and a 1.0 ohm sensing resistor (R 1 ), then loop gain for resistance compensation should be about (1.0+0.9)/1.0=1.9. Since feedback control  13  has a proportional gain of about 1.5 (15k/10k), and voltage device  5  has a gain of about 1.1, adder circuit  14  should have a gain of about 1.9/1.5/1.1=1.15 for signal  9 M. Since resistor  14 B is 27 kilohm, resistor  14 F should be set to approximately 23.5 kilohm (for a gain of 27/23.5=1.15). 
     FIG. 12  illustrates how secondary current error due to 40 Hz. exciting current can be partially corrected. The exciting current error can be estimated and subtracted from a secondary current signal to produce a more accurate secondary current signal (or an inverted exciting current error can be added to a secondary current signal). One simple way to do this is to estimate an inverted exciting current error to be a 40 Hz. sinusoid proportional to signal  9 F with a fixed phase angle change (this method does not correct for errors related to harmonic components of the exciting current error). A phase shift circuit  24  can be embodied as a bandpass filter similar to 40 Hz. bandpass filter  21  (detuned somewhat to cause the proper phase shift). Signal  9 N is then approximately proportional to an inverted 40 Hz. component of exciting current (as shown in  FIG. 9N ). A signal adder  25  (similar to signal adder  14  of  FIG. 8 ) is then used to adjust the magnitude of signal  9 N and add it to signal  9 C, thereby calculating a corrected signal V 5  that is more accurately proportional to primary current J 1  than is signal  9 C (corrected signal V 5  is a voltage signal that has almost the same waveform as primary current J 1 , which is shown in  FIG. 9A ). 
     FIG. 13  shows an alternate embodiment of the invention utilizing digital electronic calculating means for a control circuit  11 B. An analog-to-digital converter circuit  41  converts signal  9 B into a digital signal for processing by a digital signal processor  42 . If sense winding  10  is optionally included for better flux control, then analog-to-digital converter circuit  41  also converts signal  9 P into a digital signal for processing by a digital signal processor  42 . 
   Digital signal processor  42  digitally performs functions similar to the function blocks shown in  FIG. 10  for control circuit  11 A. A digital-to-analog circuit  43  converts a digital output signal from digital signal processor  42  and generates signal  9 L, which is an analog signal similar to signal  9 L in embodiments previously discussed. In this configuration voltage V 1  may be used as an output signal, or a voltage signal V 6  may be generated by control circuit  11 B. Voltage signal V 6  may include correction for exciting current error by programming digital signal processor  42  to include calculations similar to those described for  FIG. 12 . 
   Similar to  FIG. 10 , the embodiment shown in  FIG. 13  is well-suited to measuring a primary current having any combination of a-c and d-c components, including a primary current having one or more frequency components faster than the predetermined frequency. 
     FIG. 14  shows an alternate embodiment of the invention utilizing a combination of digital electronic calculating means and analog electronic calculating means for a control circuit  11 C. The operation is similar to  FIG. 10 , except that a microcontroller circuit  26  performs the functions of several function blocks shown in  FIG. 10 . These function blocks include square-wave oscillator  19 , frequency divider  20 , biphase amplifier  16 , lowpass filter  17 , and signal integrator  18 . 
   Microcontroller  26  receives signal  9 G from 80 Hz. bandpass filter  15 . An analog-to-digital converter integral to microcontroller  26  converts signal  9 G into a digital signal for processing by microcontroller  26 . Output signal  9 U may be a pulse-width modulated output signal having an average value approximately correlating to the sum of signal  9 J and signal  9 K in  FIG. 10 . Lowpass filter  27  then filters signal  9 U so that signal  9 W is approximately equal to the sum of signal  9 J and signal  9 K in  FIG. 10 . Signal  9 V is a 40 Hz. square wave similar to signal  9 E of  FIG. 10 . Bandpass filter  28  smooths signal  9 V into a sine wave to minimize noise. As in previous embodiments, the zero-flux controls associated with sense winding  10  are optional (these are shown with dashed lines). 
   Level shifter circuits may be required at the input and output of microcontroller  26  if the input and output of microcontroller  26  are not suitable for receiving and producing bipolar signals. 
   Similar to  FIG. 10  and  FIG. 13 , the embodiment shown in  FIG. 14  is well-suited to measuring a primary current having any combination of a-c and d-c components, including a primary current having one or more frequency components faster than the predetermined frequency. 
     FIG. 15  shows a simplified embodiment of the invention operating in an open-loop mode. A control circuit  11 D serves primarily as a calculating means. Operation of  FIG. 15  is similar to  FIG. 7 , except that (referring to  FIG. 7 ) the feedback control loop has been opened by disconnecting signals  9 J and  9 K from signal adder  14 . (Signal adder  14  is then no longer necessary and signal  9 L becomes the same as signal  9 F). In this configuration, the output of lowpass filter  17  (signal  9 J or voltage V 7 ) becomes a measure of the average magnetomotive force caused by primary current J 1 . Since average magnetomotive force is proportional to the d-c current component of current J 1 , voltage V 7  may also be used as a measure of d-c current in conductor  4 . 
   In addition to measuring d-c current, the configuration of  FIG. 15  may be used to measure currents having alternating current components with frequencies slower than the predetermined frequency for excitation (which is 40 Hz. in  FIG. 15 ). The practical frequency limit is dependent on properties of magnetic core  1 , response characteristics of control circuit  11 D, and, of course, the excitation frequency utilized. 
     FIG. 16  shows a typical response curve for the configuration shown in  FIG. 15 . For this curve, current J 1  is limited to d-c current, which is proportional to average magnetomotive force F 1  experienced by magnetic core  1 . Average magnetomotive force F 1  varies linearly from negative values to positive values along the horizontal axis. The magnitude of voltage V 7  is approximately proportional to the magnitude of the second harmonic component of exciting current. The polarity of voltage V 7  correlates with the phase angle of the second harmonic component of exciting current. 
     FIG. 16  graphically shows a known relationship between magnetomotive force F 1  and a difference of symmetry between excitation waveforms. In this case, the known relationship involves the magnitude and phase angle of a second harmonic component of exciting current when excitation voltage is symmetrical. 
   In the range between the positive and negative peaks of the response curve (shown in  FIG. 16 ), the response curve is a known correlation between the average value of magnetomotive force and the magnitude of the second harmonic component of exciting current. Within this range, by utilizing the known correlation, the average magnitude of magnetomotive force (or d-c current) may be determined by measuring the magnitude of the second harmonic component of exciting current. 
   Still referring to  FIG. 16 , the response curve also indicates the polarity of the average magnetomotive force, the polarity of voltage V 7  being derived from a known correlation to the phase angle of the second harmonic component of exciting current. Therefore, it may be said that the polarity of the average value of magnetomotive force has a known correlation to the phase angle of the second harmonic component of the exciting current. By utilizing this known correlation, the polarity of average magnetomotive force (or d-c current) may be determined by measuring the phase angle of the second harmonic component of exciting current. 
     FIG. 17  shows an alternate embodiment intended to generalize some aspects of the invention.  FIG. 17  illustrates how current sources may be used in place of voltage sources, how an exciting means may utilize a dedicated winding separate from a control winding, how the invention may be applied to magnetic bodies other than magnetic cores of current transformers, and how a general-purpose Proportional plus Integral plus Derivative (PID) controller may be utilized with the invention. 
   In  FIG. 17 , a magnetic body  1 B is magnetically coupled to four windings: a primary winding  4 B, a secondary winding  4 C, an excitation winding  2 B, and a control winding  2 C. Winding  4 B conducts a current J 3  as a result of being connected to a related system. Winding  4 C conducts a current J 6  also as a result of being connected to a related system. The difference in amp-turns of currents J 3  and J 6  cause magnetic core  1 B to experience an average magnetomotive force that may not equal zero. The present invention utilizes excitation winding  2 C and control winding  2 D to control the average magnetomotive force so that the operation of magnetic core  1 B is optimized for the related system. The principles described below are applicable for any number of windings coupled to the magnetic core (not just two windings ( 4 B and  4 C) as shown). 
   In  FIG. 17 , the feedback components are not required if magnetomotive force is only to be measured (but not controlled). Feedback components that may be deleted are indicated with dashed lines, and include a PID controller  55 , an operational amplifier  6 B, a resistor  51 , and a control winding  2 C. If these components are deleted, then operation is almost identical to  FIG. 15 , except that the excitation means utilizes a current source (rather than a voltage source), and it is the dissymmetry of the excitation voltage that is measured to determine magnetomotive force (rather than dissymmetry of exciting current). 
   A control circuit  11 E is a calculating means, and is similar to previous control circuits, except that it may utilize a general-purpose PID controller  55 , and may have two output signals instead of one. PID controller  55  receives signal  53  as an information signal containing information about the average value of magnetomotive force. PID controller  55  may be configured to maintain magnetomotive force at a fixed setpoint, or it may have an input signal  54  from a separate system and operate with a variable setpoint. 
   The configuration of  FIG. 17  utilizes two current sources to control the average magnetomotive force experienced by magnetic body  1 B. An operational amplifier  6 A and resistor  50  are configured as a current source and are part of an exciting means. Control circuit  11 E provides a symmetrical oscillating voltage signal  52 , which is converted to a symmetrical oscillating current J 4  by op amp  6 A. This results in a voltage V 8 , which is an input signal to control circuit  11 E (a direct connection to the output of op amp  6 A is a sensing means for sensing voltage V 8 ). The waveform symmetry of voltage V 8  is dependent on the average magnitude and polarity of magnetomotive force. Voltage V 8  is the sum of the excitation voltage and voltage drops associated with exciting current flowing through stray impedances. Since the stray impedances are generally linear (and since the exciting current is symmetrical), the voltage drops have symmetrical waveforms and do not contribute to the dissymmetry of the waveform of voltage V 8 . Therefore, any even harmonic components present in voltage V 8  are due to the excitation voltage. Control circuit  11 E processes this unsymmetrical signal in a similar manner as previously describe for other control circuits. 
   PID controller  55  controls a second current source comprising an operational amplifier  6 B and resistor  51 . A voltage signal  56  from PID controller  55  is converted to a current J 5  that is proportional to voltage signal  56 . Current J 5  directly adds or subtracts from the magnetomotive force to maintain the average magnetomotive force at the setpoint of PID controller  55 . 
   Finally, for purposes of clarifying the claims, the word “apparatus” has its usual meaning: a set of material or equipment designed for a particular use. 
   While several embodiments have been described and illustrated, there are many other embodiments possible that will be apparent to those skilled in the art. It is not the intent of this disclosure to limit the invention to the embodiments that have been illustrated. The components and configurations utilized in this disclosure are intended to be illustrative only, and are not intended to limit the scope of the appended claims. 
   While only certain preferred features of the invention have been shown by way of illustration, many modifications and changes will occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.