NIST Technical Note Transformer-Like Devices for High-Accuracy AC Current Measurements. T. M. Souders

Transcription

1 T Technical ote 1473 Transforer-Like Devices for High-Accuracy AC Current Measureents T. M. ouders

2 T Technical ote 1473 Transforer-Like Devices for High-Accuracy AC Current Measureents T. M. ouders June 2008 U.. Departent of Coerce Carlos M. Gutierrez, ecretary ational nstitute of tandards and Technology Jaes M. Turner, Deputy Director

3 Certain coercial entities, equipent, or aterials ay be identified in this docuent in order to describe an experiental procedure or concept adequately. uch identification is not intended to iply recoendation or endorseent by the ational nstitute of tandards and Technology, nor is it intended to iply that the entities, aterials, or equipent are necessarily the best available for the purpose. ational nstitute of tandards and Technology pecial ublication 1473 atl. nst. tand. Technol. pec. ubl. 1473, 62 pages (June 2008) CODE: UE2

4 Table of Contents FOREWORD.... ix ABTRACT 1 1. TRODUCTO The deal Transforer Transforer Error and its Representation Other Error ources otation HYCAL TERRETATO OF TRAFORMER ERROR THE CURRET COMARATOR The Copensated Current Coparator MAGETC DEG EQUATO AD THE MEAUREMET OF MAGETC ROERTE Hysteresis, ereability and Core Loss Magnetic Flux Density and aturation Flux Density Magnetizing pedance and Magnetizing pedance per Turn-quared Detection ensitivity for Current Coparators, and Tuned Detection Core Deagnetization WDG TECHQUE AD THE ETMATO OF LEAKAGE MEDACE uber of Turns and Wire ize Winding Layout The ingle-turn roble Turns Counting AVE AD ACTVE CORRECTO TECHQUE assive Two-tage Current Transforers Active Two-tage Transforers Active Current-Coparator Correction Circuits Construction of Two-tage Transforers and Copensated Current Coparators Ratio Cascading MAGETC ERROR: T MEAUREMET AD MTGATO. 28 iii

5 7.1 ource of Magnetic Error Test Method for Magnetic Error Magnetic hielding to Reduce Magnetic Error CALCULATO AD MAAGEMET OF CAACTVE ERROR Evaluation of ign Factor, k C Calculating Capacitive Errors: Exaples Triing of Capacitive Errors in Transforers or Copensated Current Coparators Additional Coents on Capacitive Errors CURRET TRAFORMER TETG Transforer Testing Using a tandard Transforer Transforer Testing Using a Copensated Current Coparator econdary Feed with a Copensated Current Coparator Testing Transforers with Ratios Less Than Unity Measuring the Test Transforer Burden Calibration Methods and Deterination of Uncertainties REFERECE 50 iv

6 List of Figures Figure 1.1 Current Measureent Using Voltage-Based nstruentation. 1 Figure 1.2 The deal Transforer. 2 Figure 1.3 Equivalent Circuit of Two-Winding Transforer... 3 Figure 1.4 otation of Cores and Windings. 4 Figure 2.1 Current Transforer with quare-cross-ection Toroidal Core (1/2 shown) 5 Figure 3.1a iple Current Coparator. 7 Figure 3.1b cheatic Representation of iple Current Coparator.. 8 Figure 3.2 Equivalent Circuit of a iple Current Coparator 8 Figure 3.3 Copensated Current Coparator.. 10 Figure 3.4 Copensated Current Coparator: Equivalent Circuit.. 11 Figure 4.1 Hysteresis Loop.. 12 Figure 4.2 Test etup to Measure Magnetizing pedance per Turn-quared (left), and its Equivalent Circuit (right). 14 Figure 4.3 Tuned Detection Circuit (left), and Equivalent Circuit (right) Figure 4.4 Core Deagnetization. 16 Figure 5.1 eries-arallel Arrangeent for riary Winding (one exaple). 18 Figure 5.2 olution for ingle-turn roble 19 Figure 5.3 Turns Counting Circuit. 19 Figure 6.1 Two-tage Current Transforer with eparate Burdens 20 Figure 6.2 Two-tage Transforer with eparate Burdens: Equivalent Circuit. 21 Figure 6.3 Two-tage Transforer-caled Resistor 22 Figure 6.4a,b Two-tage Transforer with ingle Burden: Equivalent Circuits. 22 Figure 6.4c Two-tage Transforer with ingle Burden: Final Equivalent Circuit. 23 v

7 Figure 6.5 Two-tage Transforer with Feedback Aplifier. 24 Figure 6.6 Aplifier-Aided Two-tage Transforer.. 24 Figure 6.7 elf-balancing Current Coparator 25 Figure 6.8 elf-balancing Current Coparator: Equivalent Circuit 26 Figure 6.9 Miljanic-o-Moore Circuit. 26 Figure 6.10 Miljanic-o-Moore Equivalent Circuit Figure 6.11 Construction of Two-tage Transforers and Copensated Current Coparators.. 28 Figure 6.12 Cascading Two-tage and Aplifier-Aided Two-tage Transforers. 28 Figure 7.1 Unequal ensitivities Due to Magnetic Error. 29 Figure 7.2 Manifestation of Magnetic Error. 29 Figure 7.3 ource of Magnetic Error 30 Figure 7.4 Test Method for Assessing Magnetic Error 31 Figure 7.5 Use of Magnetic hield to Reduce Magnetic Error 32 Figure 7.6 Magnetic hield ½ of hield hown to illustrate Cross ection. 32 Figure 7.7 Eddy Current hield ½ of hield hown to illustrate Cross ection.. 33 Figure 8.1 Figure 8.2 Capacitive Current in econdary Winding of a Transforer or Copensated Current Coparator, with a hield Connected to the Defined Terinal. 36 Capacitive Current in econdary Winding of a Transforer or Copensated Current Coparator, with a Grounded hield and Marked Terinal at Virtual Ground 37 Figure 8.3 Turn-to-Turn Capacitance of econdary Winding. 38 Figure 8.4 Capacitive Current in riary Winding of a Transforer or Copensated Current Coparator, with a Grounded hield and Marked Terinal at Virtual Ground 39 Figure 8.5 Turn-to-Turn Capacitance of riary Winding. 40 vi

8 Figure 8.6 Triing Capacitive Error in a Two-tage Transforer or Copensated Current Coparator. 41 Figure 9.1 Transforer Testing with a Transforer-Like tandard 43 Figure 9.2 Figure 9.3 Test et Based on Measureent of Error Current Through a all Resistance. 44 Test et Based on Measureent of Error Current in a Current Coparator Winding.. 44 Figure 9.4 Transforer Testing with a Copensated Current Coparator 45 Figure 9.5 Figure 9.6 Transforer Testing with Copensated Current Coparator Using econdary Feed 47 pecial urpose Copensated Current Coparator for Ratios Less Than Unity. 48 Figure 10.1 Calibration of the 1/1 Base Ratio of a Transforer 49 vii

9 List of Tables Table 5.1 Approxiate Current Carrying Capacity of elected Wire izes. 17 Table 7.1 Lowest Achievable Magnetic Error for Different Configurations 34 Table 8.1 Evaluation of ign Factor, k C 35 viii

10 FOREWORD Between 1967 and 1975, the author conducted the calibration services for instruent current transforers at the ational Bureau of tandards (now ational nstitute of tandards and Technology), and upgraded the facilities used in those services. n addition, he developed and ipleented a new easureent service for low-value ac resistors. Much of the aterial included in this Technical ote derives fro work during that period. Rearkably, the field has not changed draatically over the intervening years. The period fro about 1958 to 1970 saw renewed interest in transforer-like devices as solutions to a variety of electrical easureent probles, and developent progras were undertaken at a nuber of national laboratories and universities. Many of these led to new types of devices having vastly iproved accuracy, aking it possible to scale and easure ipedances, ac voltage and current, and power and energy, with unprecedented accuracy. During this period, an ongoing collaboration between.. Miljanic of the nstitut ikola Tesla in Belgrade, Yugoslavia and. L. Kusters and W. J. M. Moore of the ational Research Council (RC) in Ottawa, Canada, led to the developent of the odern current coparator. This represented a new class of transforerlike devices with rearkably increased accuracy over conventional current transforers. O. etersons, also with RC during part of that tie, extended and applied the work of that group, particularly to the field of high voltage easureents. Many of these developents were available to the author as he began the design of the new easureent services at B. Around 1968 etersons left RC to take a position at B and the author was fortunate to have hi as a entor during the first few years of his career. Many of the ideas and approaches set forth in this docuent, especially regarding agnetic shielding and the use of equivalent circuits, can be traced to his influence. The work was perfored under the supervision of B. L. Dunfee, who provided a supportive, nurturing work environent for which the author has ever since been thankful. n addition, colleagues D. Flach and R. Kahler were partners during part of this work, and contributed their own diligence and insights. Of course, any errors or oissions in this docuent are solely the author s responsibility. ix

11 TRAFORMER-LKE DEVCE FOR HGH-ACCURACY AC CURRET MEAUREMET T.M. ouders ational nstitute of tandards and Technology, Gaithersburg, MD ABTRACT A theoretical and practical fraework is presented to aid in the design, fabrication, and testing of transforer-like devices for use in high-accuracy ac current etering applications. Current transforers, two-stage current transforers, and current coparators are discussed, as well as related devices that use passive and active error correction techniques. Transforer theory is developed in ters of siple electroagnetic theory and practical equivalent circuits. Magnetic design equations are presented and the easureent of relevant agnetic properties is discussed. ources of error and their itigation are covered in detail, including errors caused by agnetizing currents, winding and core inhoogenieties (so-called agnetic errors), and circulating capacitive currents. Calibration ethods and current transforer testing are also covered. KEY WORD: ac current easureent; current coparators; current transforers; current transforer testing; error sources; equivalent circuits; agnetic shielding; transforer design 1

12 1. TRODUCTO nstruent current transforers are used to scale ac currents to levels that are ost appropriate for easureent. Today, ost odern electronic instruentation is voltage-based: Current is easured by first passing it through an ipedance of known value, and the resulting voltage drop becoes the quantity that is actually easured (see fig. 1). Most often the ipedance is a low inductance four-terinal resistor, although utual inductors and even capacitors are occasionally used for ac easureents. To achieve the highest accuracy, the ipedance value is generally selected to give an rs voltage level in the range of 0.1 V to 1 V. However at current levels greater than about 1 A, it becoes increasingly difficult to use resistors because of probles of power dissipation and residual inductance. For exaple, if the voltage drop is aintained at 1 V, then one watt of power dissipation is required for every apere of easured current. Furtherore, the residual inductance associated with any resistor design causes phase errors (between input current and output voltage) that becoe unacceptably large at high currents (i.e., low resistance values) and high frequencies. Fortunately, pre-scaling the current to be easured can iniize these probles. Figure 1.1. Current Measureent Using Voltage-Based nstruentation Current transforers are typically used for this purpose since they are capable of reasonably high ratio accuracies with relatively low power dissipation over a wide range of current ratios. The current flowing in a load (or burden, B) connected to the secondary winding of a current transforer is noinally equal to the current in the priary or driven winding, ties the inverse turns ratio. Consequently, under the conditions given above, a current transforer having an inverse turns ratio of will reduce the power dissipation in the resistor by a factor of 1/ for a fixed voltage. Even higher accuracies are achievable using feedback aplifier techniques or transforers with two or ore stages. For the special case in which the current in question is being copared with a standard, known current of the sae frequency, then current coparators can be used. This technical note presents a theoretical and practical fraework to aid in the design, fabrication and testing of current transforers and current coparators for use in high accuracy etering applications. 1.1 The deal Transforer Real transforers are designed to approxiate the properties of an ideal transforer as represented in fig The ideal transforer is a device having two agnetically coupled but galvanically isolated windings designated priary and secondary, with and turns 2

13 respectively. The windings theselves of the ideal transforer have zero ipedance, and are perfectly coupled, so that the ratios of the voltages (or currents) appearing at the terinals are exactly equal to the turns ratio (or inverse turns ratio) as indicated in the figure. When ipedance, B, is connected to the secondary winding, it follows then that the ipedance as easured fro the priary terinals is ultiplied by the square of the turns ratio. Figure 1.2. The deal Transforer 1.2 Transforer Error and its Representation The error,, of a current transforer is a coplex quantity that expresses the degree to which the true priary-to-secondary current ratio differs fro the inverse turns ratio: ' 1 1 j, (1.1) where and are the in-phase and quadrature error coponents, respectively. The notation is used to indicate that the actual secondary current differs fro the idea secondary current,, that is shown in fig n real transforers, the driving voltage that supports the secondary current is induced by the ac flux in the agnetic core linking the windings. However, to sustain the ac flux a agnetizing current ust link the core; i.e., the net difference between the priary and secondary apereturns ust not be zero. By definition though, the net apere-turns of an ideal transforer is zero ( - = 0) as stated above; consequently it is this residual agnetizing current that is responsible for the ajor error associated with current transforers. By using toroidal cores with high agnetic pereability, the agnetizing current can usually be kept sall, but it cannot be reduced to zero. The equivalent circuit shown in fig. 1.3 can be used to accurately represent this error source, along with other characteristics of real transforers such as finite winding ipedances. Here and represent the so-called leakage ipedances that characterize the priary and secondary windings, and is the agnetizing ipedance, a characteristic of the 3

14 core and winding that ultiately deterines the size of the agnetizing current,. ote that the agnetizing current is shunted away and current is the actual current delivered to the secondary burden. n practical designs, is large copared to and B, so that is sall copared to, and the resulting transforer error is sall. These concepts will be developed ore fully in subsequent chapters, and the quantities and will be related to the underlying physics involved. Figure 1.3. Equivalent Circuit of Two-Winding Transforer For now, we can solve the circuit equations fro fig. 1.3 to obtain the transforer error as defined in (1.1) above, in ters of the circuit paraeters. Thus we have ' 1 B 1 (for B ) B, (1.2) 1 and cobining (1.2) with (1.1) we get the transforer error as B. (1.3) 1.3 Other Error ources While the agnetizing current is the predoinant source of error in a siple current transforer, it is not the only source. o-called agnetic errors arise when the windings link the core unequally (see chapter 7), and capacitive errors arise when capacitance between or across windings shunts part of the current away fro the core (see chapter 8). For current coparators and ultistage current transforers, these becoe the doinant sources of error. ubsequent chapters will explore these error sources in depth, and will present approaches for their itigation. ote however that the type of equivalent circuit shown in fig. 1.3 cannot easily represent either of these error sources, and it will only be used to represent errors caused by agnetizing currents. 4

15 1.4 otation Fig. 1.4 shows the notation that will be used throughout this docuent to indicate the relative placeent of windings and cores, and the polarity of windings designated by dots placed at one end of each winding. There are two governing rules: 1. Only windings shown above a given core link that core. 2. olarity: For all currents entering corresponding terinals of windings linking a coon core, the direction of flux induced in the core is the sae. This causes all characteristic ipedances to be positive. Figure 1.4. otation of Cores and Windings Therefore, in fig. 1.4, the priary, secondary and tertiary windings all link core 2, but only the priary and secondary windings link core HYCAL TERRETATO OF TRAFORMER ERROR Two laws of electroagnetic theory govern the fundaental principles of current transforers and current coparators: Faraday s Law of nduction and Apere s Circuital Law. Together with the coon forula for the inductance of a winding on a toroidal core of square cross section, Faraday s law can be used to calculate the error of a current transforer caused by agnetizing current. Fig. 2.1 represents a current transforer consisting of a agnetic core wound with a priary winding of turns and a secondary winding of turns connected to a burden. The supplied priary current enters the arked priary terinal, and in accordance with Le Chatelier s principle of least action, the resulting secondary current leaves the arked secondary terinal, thus iniizing the net flux in the core and the net apere-turns linking the core. f we think of the net apere-turns as a net current, i, flowing in the secondary winding, then we have ' i ( i i ) /. (2.1) n a current transforer, i is called the agnetizing current. 5

16 Figure 2.1. Current Transforer with quare-cross-ection Toroidal Core (1/2 shown) Fro Faraday s law, the voltage induced in the secondary winding is given by v E dl d L dt di dt, (2.2) where v E dl L is the induced voltage (V), is the electric field intensity along path of the winding (V/), is an eleent of length of the path of the winding (), is the agnetic flux in the core (Wb), and is the inductance of winding (H). The standard units of easureent are shown in parentheses, and the lower case notation used in (2.1) and (2.2) denotes tie-doain variables. Transforing to the frequency doain, (2.2) becoes V j L. (2.3) n a current transforer, the induced secondary winding voltage, V, supports the voltage drop across the burden ipedance and the secondary winding leakage ipedance,. (The leakage ipedance of a winding consists of the winding resistance, and the coponent of inductive reactance whose flux does not link other windings). Therefore, V ' ( ). (2.4) B Cobining (2.3) and (2.4) gives the following expression for the agnetizing current: 6

17 ( jl ' B (2.5) ) Fro (2.5), the transforer ratio is then given by ' ( / ) ( )( / ) B 1, (2.6) ' ' ' jl and coparing (2.6) with (1.1) we see that jl B. (2.7) f we call ipedance jl the agnetizing ipedance,, then we arrive at the sae error expression that is given by the equivalent circuit, i.e., (1.3). Furtherore, we see that the agnetizing ipedance can be calculated fro the siple forula for the inductance of a winding on a toroidal core as shown in fig. 2.1: L 2 k0 A 2 R ( R w), (2.8) where L is the inductance of the winding (H), k is the relative pereability of the core aterial (diensionless), is the pereability of free space (410-7 Wb/A-), is the nuber of turns of the secondary winding, A is ithe effective cross sectional core area ( 2 ), R is the ean radius of the core (), and w is the width and height of the core (). n practice, the agnetizing ipedance is not a pure inductance but includes a resistive coponent that reflects the losses in the core aterial. At low frequencies, the losses are typically sall and (2.7) gives a good approxiation for the error. Fro a practical standpoint, the forula given in (2.8) can be used for ost coon core configurations since it is reasonably accurate for any rectangular core cross section in which the ratio of height to width is at least 0.5. Before oving on to the next section, note that the transforer error given in (2.7) is only dependent on the paraeters of the secondary circuit. 3. THE CURRET COMARATOR n any applications, it is useful to be able to accurately copare a current with a reference current when the two currents are of different agnitudes. uch needs arise, for exaple, in 7

18 any types of bridge circuits as well as in apparatus for testing current transforers. The siple current coparator shown in figs. 3.1a and 3.1b is often used in these applications. Historically, the current coparator was described as early as 1917 by Baker [27], and was later rediscovered and iproved upon by Obradovic, Miljanic and piridonovic in 1957 [25] and by Kusters and Moore in 1961 [23]. ubsequent collaborations between Miljanic, Kusters and Moore led to any further developents and iproveents (see [4] for a thorough description of this work). The currents being copared are carried by two ratio windings as shown, with opposing polarities, and a third detection winding is used to indicate when apere-turn balance is achieved, i.e, when the coplex ratio of the two currents is exactly equal to the inverse turns ratio of the coparator. Under this condition, the flux in the core is zero, and therefore no voltage is induced in the detection winding. n soe applications, one of the currents is adjusted by known aounts to bring about the null condition, while in other applications the nuber of turns in one or both windings is adjusted for the sae purpose. ince there is no flux in the core at balance, there are no voltages induced in the windings and no agnetizing current is present; consequently, no power is transferred fro priary to secondary circuit, as is the case with current transforers. nstead, the power dissipated in the secondary circuit of a siple current coparator is totally supplied by the source of the secondary current. Figure 3.1a. iple Current Coparator These principles are ebodied in the equivalent circuit shown in fig This circuit is siilar to the current transforer equivalent circuit of fig. 1.3 with the addition of another ideal transforer to represent the placeent and action of the detection winding. ince the detection winding is used to detect the presence of a agnetizing current, the upper winding of the second ideal transforer is connected across the agnetizing ipedance, and the turns ratio of this transforer represents the ratio of the secondary winding to the detection winding. When the voltage at the detector is zero, the voltage across the agnetizing ipedance, and consequently the agnetizing current, ust also be zero. Under these conditions, the actual secondary current,, equals the ideal secondary current,. As noted earlier however, agnetic and capacitive errors also contribute to the overall accuracy of a current coparator, and these error sources are considered in detail in subsequent chapters. 8

19 Figure 3.1b. cheatic Representation of iple Current Coparator Figure 3.2. Equivalent Circuit of a iple Current Coparator hysically, the current coparator is often thought of as an ebodient of Apere s Circuital Law: The line integral of the agnetic field intensity, H, around a closed path is equal to the su of the currents that are enclosed by that path. Although the rationale is a bit tenuous, it proceeds as follows. f the path taken is that of the agnetic core, and the currents are the priary- and secondary-winding currents flowing in opposition through and turns, respectively, this gives where dl s H d, (3.1) is an eleent of length of the path () and indicates suation over any surface enclosed by the line integral. n a current coparator, the line integral is estiated via a detection winding that densely and uniforly covers the core (unlike that illustrated in fig. 3.1a, where, for clarity, the detection winding is shown covering only a portion of the core). 9

20 f it is assued that the flux density, B = k 0 H, is constant over the core s cross section, then (3.1) can be written in ters of agnetic flux as where k 0 A 1 k 0 A is the relative pereability, is the pereability of free space, is the cross sectional area of core, and s the agnetic flux. dl, (3.2) For regular toroidal cores and uniforly distributed windings, we can ake the further siplifying assuption that the flux is constant over the path of the core, giving where R is the radius of the toroidal core. 2 R k A 0, (3.3) Meanwhile, the detection winding voltage can be expressed in ters of the agnetic flux in the core fro Faraday s law as d vd ED dl D. (3.4) dt (ote that the line integral in (3.4) follows the path of the detection winding around and around the core cross-section D ties. Although this path eventually traverses the core, it is not the sae as the line integral of (3.1) which for siplicity is assued to follow the core path itself. f the sae path were used in both, a less restrictive arguent could be ade, however with the expense of greater coplexity.) For sinusoidal signals (3.4) becoes V j, (3.5) D D and cobining (3.5) with the frequency-doain counterpart of (3.3), we get V D k j 0 A 2R D. (3.6) Therefore, the detection winding voltage is directly proportional to the net apere-turns that link the core, i.e., that pass through the core window. The condition of apere-turn balance that occurs in siple current coparators when the priary and secondary apere-turns are equal and opposite, is indicated by a null voltage at the detection winding. 10

21 Fro the equivalent circuit of fig. 3.2, we can see that the open-circuit voltage at the detector is given by D ' D ' D ' D VD ( ) ( ) 2. (3.7) ubstituting into (3.7) the forula for agnetizing ipedance derived previously (see (2.8)) gives the sae expression for detection winding voltage as shown in (3.6). Therefore the equivalent circuit is consistent with the physical interpretation given above. The quantity 2 D / in (3.7) is called the sensitivity of the detection winding. Under the various assuptions ade above, the sensitivity to apere-turns in the priary winding is the sae as the sensitivity to apere-turns in the secondary winding, as (3.7) and (3.6) suggest; and in fact the accuracy of the current coparator depends on this equality. However, so-called agnetic errors can arise when the assuptions are not strictly valid, and the result is that the sensitivity is soewhat different for the priary and secondary windings. The subject of agnetic error and its itigation is treated in chapter The Copensated Current Coparator One drawback of the siple current coparator is that the leakage ipedances of both ratio windings can be significant loads for the respective current sources. f the current coparator is being used to calibrate a current transforer for exaple, the source of secondary current is actually the secondary winding of the transforer under test, and the secondary leakage ipedance of the current coparator adds a significant burden to the test transforer that will affect its error. Figure 3.3 Copensated Current Coparator The so-called copensated current coparator [11] shown in fig. 3.3 iniizes this proble for the secondary circuit. This circuit has an additional core that creates a transforer stage capable of transferring power across the core fro the priary to secondary circuit. n operation, a copensation winding added to the detection core carries the agnetizing current, 1, of the transforer stage as shown, so that the voltage, V, of the secondary current source at detector balance is given by: 11

22 V 1 C, (3.8) where C is the leakage ipedance of the copensation winding. This is ore apparent fro the equivalent circuit shown in fig When the secondary current source is adjusted to achieve a null on the detector, no current flows through the agnetizing ipedance, 2, of the detection core. Therefore, current 1 that flows through the leakage ipedance, C, of the copensation winding, exactly equals the agnetizing current 1, and the secondary current exactly equals the ideal secondary current,. Figure 3.4 Copensated Current Coparator: Equivalent Circuit At detector balance, the equivalent load seen by the secondary current source is: L V 1 C ( B ) C, (3.9) 1 which is norally quite sall as copared to a load of which would exist for a siple, uncopensated current coparator. Applications of the copensated current coparator will be discussed in chapters 6 and MAGETC DEG EQUATO AD THE MEAUREMET OF MAGETC ROERTE 4.1 Hysteresis, ereability and Core Loss The so-called hysteresis loop of a agnetic core reveals several iportant characteristics that ust be considered when selecting the cores to be used in transforers or current coparators. The agnetic field intensity, H, in a agnetic core is a eoryless linear function of the current linking the core. However, the agnetic flux density, B, is only linearly proportional to the current when the core is a non-agnetic aterial, i.e., when the pereability is that of free space, 0. Otherwise, depending on the type of agnetic aterial used, the agnetic flux density is a nonlinear and non-unique function of current (or agnetic field intensity). 12

23 Figure 4.1 Hysteresis Loop These relationships are illustrated with the B/H curve or hysteresis loop shown in fig. 4.1, which is typical of the agnetic cores used in ost current transforers and current coparators. By definition, the slope of the curve, i.e., B/H, is the agnetic pereability, k 0, where k is the relative pereability. As the current increases fro zero, the pereability has an initial value that increases as the current gets larger, until a point at which it begins to decrease, ultiately reaching a value of 0 as the current becoes very large. f the current is then cycled back through zero to a large negative value, and so on for a periodic signal, the eory effect of the hysteresis loop is anifested. everal iportant points are illustrated here. First, since the pereability is a nonlinear function of current, we can expect the resulting agnetizing current of a transforer (as described in (2.5)) to contain haronic coponents even when the priary current is strictly a single tone sinusoid. However, because of the syetry of the hysteresis loop, only odd haronics will norally appear. (f the core has residual dc agnetization though, the loop will becoe asyetric and even haronics will also be generated.) econd, the relative size of the agnetizing current, and hence the error, varies depending on the flux density in the core (see section 4.2). Third, if the peak flux density becoes too large, the core saturates, the pereability pluets and the error increases draatically (see section 4.2). Fourth, the hysteretic property of the B/H curve suggests that energy is being used to agnetize the core, which results in core loss. n fact, for repetitive signals, e.g., sinusoids, the total energy per unit core volue per cycle is equal to the area enclosed by the loop. The resulting core loss is represented in an equivalent circuit as a parallel loss coponent of the agnetizing ipedance. Finally, the initial state of agnetization affects the overall size and syetry of the B/H curve (and the haronic content as noted above) and deagnetization ay be required to achieve the initial state of zero agnetization that is assued in fig. 4.1 (see section 4.5). 13

24 4.2 Magnetic Flux Density and aturation Flux Density n order to estiate the size of agnetizing currents in transforer designs and hence the errors that result, it is necessary to know the agnetic flux density that can be expected in the cores. For toroidal cores, the agnetic flux density ay be calculated in ters of the secondary current, secondary leakage ipedance and burden, as follows. Fro (2.2), the frequency doain relationship between secondary winding voltage, V, flux,, and flux density is given by V j j B A, (4.1) where A is the nuber of turns in the secondary winding and is the effective cross sectional area of the core. Rearranging (4.1) yields V B j, (4.2) 2 f A and cobining this with (2.4) gives the agnetic flux density as ' ( B ) B j. (4.3) 2 f A With expressed in aperes, the ipedances in ohs, and A in squared eters, B is given in tesla or Wb/ 2. ote that 1T equals 10 4 gauss, with gauss being the unit ore coonly used by anufacturers of agnetic cores. f V is assued to be the rs value of the secondary voltage and B sat is the saturation flux density, then the largest secondary voltage that can be sustained, V ax, is obtained fro (4.2) as V ax j2 f ABsat. (4.4) Magnetizing pedance and Magnetizing pedance per Turn-quared As noted in chapter 2, the principal coponent of the agnetizing ipedance,, associated with a winding on a core is the self-inductance of the winding. o again, for toroidal cores the agnetizing ipedance is given by: 2 2 k 0 A f k 0 A jl j j. (4.5) 2 R R This expression ignores the contribution of core loss, which is typically rather sall at low frequencies. 14

25 The ipedance given in (4.5) corresponds to a particular nuber of turns,, of the winding. For design purposes, it is usually ore convenient to characterize the core itself by its agnetizing ipedance per turn-squared, given by: f k A j. (4.6) R 0 2 While the expressions of (4.5) or (4.6) are critical to the selection of cores and the nuber of turns to use in a transforer design, they only provide approxiate estiates of device perforance. As noted above, they ignore contributions such as core loss, and the values for the paraeters on which they depend are usually typical values taken fro data sheets. To get a ore accurate estiate of the perforance one is likely to achieve once a core has been selected, it is certainly advisable to easure the agnetizing ipedance per turn-squared directly. uch a easureent is siple to perfor with the setup shown in fig Figure 4.2 Test etup to Measure Magnetizing pedance per Turn-quared (left), and its Equivalent Circuit (right) With this setup, two windings are placed on the core under test. The first, with 1 turns, carries the excitation current that is easured in ters of voltage V 1 across the series ac resistor. The second winding of 2 turns, produces the induced voltage, V 2. The current is adjusted to give the desired agnetic flux density test condition, which is indicated per (4.2) by voltage, V = V 2, and then the two voltages are recorded. The agnetizing ipedance per turn-squared is given as: V2R. (4.7) 12 V1 By using two windings, this approach akes it possible to easure the ipedance per turnsquared independent of the leakage ipedance of either winding. The values for 1 and 2 are selected for convenience of easureent, but soe care should be taken to distribute each uniforly around the core. 4.4 Detection ensitivity for Current Coparators, and Tuned Detection The function of a current coparator is to detect the condition of apere-turn balance aong the ratio windings. As discussed in chapter 3, apere-turn balance is sensed by a detection winding 15

26 placed on the detection core. The ain paraeter of interest in selecting the detection core and nuber of detection winding turns is the detection sensitivity, i.e., the detection winding voltage produced per apere-turn of unbalance. Referring to fig. 3.2, any apere-turn unbalance gives rise to the agnetizing current,, which flows through the agnetizing ipedance,. Therefore, the detection voltage, V D, is given by: The detection sensitivity, D-, is then: D VD. (4.8) V D D D, (4.9) 2 since is the apere-turn unbalance. For toroidal cores, we cobine (4.9) with (4.6) to yield: D f k A D j 0. (4.10) R f 4.10 is expressed in ters of winding density, i.e., the nuber of turns in the detection winding per unit length of core circuference, D T, then we have [4]: jdt k A, (4.11) D 0 where DT D /( 2R). Expressed in this way, we see that the sensitivity does not depend on the diaeter of the core, but only on the winding density and the cross sectional area of the core. As previously noted, the agnetizing ipedance is priarily inductive at lower frequencies, and so the detection sensitivity is also priarily inductive as the iaginary ter in (4.10) iplies. This akes it possible to increase the detection sensitivity by tuning the circuit with a capacitor across the detection winding as shown in fig Figure 4.3 Tuned Detection Circuit (left), and Equivalent Circuit (right) 16

27 The sensitivity is greatest when the capacitance is given by L C, (12) 2 2 ( L) D 2 where L is the inductance of the detection winding, ( k 0 D A) /(2 R), and the leakage ipedance, D, is assued to be resistive [4]. 4.5 Core Deagnetization Figure 4.4 Core Deagnetization At the beginning of this chapter, it was noted that dc agnetization of a core affects the size and shape of the B/H curve, which in turn can affect the overall error perforance of the device. To reove residual agnetization, it is necessary to raise the ac winding voltage to the point that the core saturates, and then reduce the voltage slowly to zero. This process is indicated in fig The voltage ust not be switched off before it reaches zero, since the resulting transient ay reagnetize the core. 5. WDG TECHQUE AD THE ETMATO OF LEAKAGE MEDACE There are three basic choices to be ade in designing the windings of a current transforer or current coparator: the nuber of turns, the wire size, and the winding layout. These choices directly affect the winding s agnetizing ipedance, its current carrying capacity and leakage ipedance, as well as the resulting agnetic and capacitive errors. 5.1 uber of Turns and Wire ize The first consideration when selecting the nuber of turns and wire size to use ust be the current carrying capacity required of the winding. Generally, for windings that carry little or no current such as detection, copensation or tertiary windings, self-heating is of little concern; however for the ain ratio windings it becoes paraount. While the aount of power generated in a winding is of course the square of the winding current ties the winding resistance, the teperature buildup is ore difficult to calculate since it depends on how readily heat can be reoved fro the winding. However, for ost practical cases a rule-of-thub of 17

28 240 aperes per c 2 of wire cross-section should be acceptable. This applies to devices with ultiple windings in a reasonably ventilated space aintained at roo teperature. Table 5.1 presents the approxiate current carrying capacity of several wire sizes based on this rule. Of course, a heavier wire (i.e, saller wire gage) than shown in Table 5.1 can be used, and ay be appropriate to achieve the desired resistance for the winding. Table 5.1 Approxiate Current Carrying Capacity of elected Wire izes Wire ize (AWG) Current Carrying Capacity (A) # # # # # The nuber of turns to use for a winding depends on the type of winding. For a detection winding, the required sensitivity dictates the nuber of turns needed, as discussed in section 4.4. For tertiary or copensation windings, the nuber of turns is usually selected to atch that of the secondary winding, and the wire size for these is usually selected to give the lowest resistance that can be achieved in a single-layer winding, although in soe cases ulti-layer windings ay be required. The effects of the resistances of these windings on the device error are given in chapter 6. For the secondary windings of transforer stages, the required agnetizing ipedance sets the lower bound for the nuber of turns (see sections 1.2, 3.1, 4.3 and chapter 6), but other considerations such as saturation flux density can dictate a larger nuber (see section 4.2). Of course, the nuber of turns selected for the secondary winding ust give an integer nuber of turns for the priary winding, to achieve the desired turns ratio. 5.2 Winding Layout As seen in previous chapters, the error of a transforer stage that results fro agnetizing current is only dependent on the nuber of turns of the secondary winding. Therefore, it is usually preferable to have a fixed nuber of turns for the secondary winding since this practice keeps the error constant over all available ratios. Multiple ratios are then accoodated via the priary winding. n chapter 7, we will see that winding unifority is iportant in iniizing so-called agnetic errors, so it is also iportant to distribute the turns of a winding uniforly around the core, preferably in one layer. This also akes it easier to calculate errors due to circulating capacitive currents as discussed in chapter 8. The preference for single-layer windings has two reasons: interwinding capacitance is greatly increased in ulti-layer windings, leading to larger capacitive errors; and the leakage inductance of a winding increases substantially with ultiple layers since flux between the layers does not link all of the turns. The leakage inductance is also greater in the outer winding, since the flux in the space between windings does not link the inner one at all. Therefore, it is coon practice to ake the secondary the inner winding, thus iniizing its leakage ipedance and the resulting error that it causes in conjunction with the agnetizing ipedance. The reactance of a single layer secondary winding is then usually negligible at lower frequencies. For ulti-layer windings though, the reactance can be the doinant source of leakage ipedance, especially at higher 18

29 frequencies. Also, the leakage reactance of the priary (outer) winding is relatively larger. ote that it is experientally difficult to easure the individual leakage ipedances of each winding, except for their dc resistive coponents. Therefore they ust be estiated analytically whenever it is likely that the reactances will be significant. Of course, the resistive coponents can be readily estiated fro the resistance per unit length of the wire (available fro wire tables) and the length of wire in the winding. Forulae for calculating leakage reactances can be found in [1,19]. For ulti-ratio devices, a series-parallel arrangeent of the priary winding as illustrated in fig. 5.1 can be used to advantage. The winding is divided into individual sections of Q turns each, with the sections wound side-by-side such that the core circuference is filled by the sections. To achieve the lowest available ratio, /, all sections are connected in series. Higher ratios are available by connecting the sections in series-parallel cobinations as shown in the figure. For the exaple in the figure, is 8 and Q is 30. f the secondary winding has 240 turns, then the arrangeent in the figure can give four ratios: 1/1 (240/240), 2/1 (240/120), 4/1 (240/60), and 8/1 (240/30). ote that for a fixed secondary current level, the priary current increases in proportion to the turns ratio, but the current flowing in each of the sections is constant regardless of ratio and the power dissipation in the winding is also constant. Furtherore, the apere-turn current distribution in the winding does not change fro ratio to ratio, so that any agnetic error (see chapter 7) that exists should be independent of ratio as well. This ethod iniizes the total nuber of turns required in the priary winding, and requires wire of only one size. n addition, the priary leakage ipedance siply scales inversely with the square of the ratio. The drawbacks of the series-parallel approach are that the available ratios are liited, and procedures used to ake the necessary connections are ore coplicated than is otherwise required. Figure 5.1 eries-arallel Arrangeent for riary Winding (one exaple) 5.3 The ingle-turn roble As an -turn winding progresses around the circuference of the core, it not only links the flux in the core ties, but it also links once any flux in the window of the core. Flux in this region is extraneous and will induce an additional voltage in the winding that is unwanted. uch flux often represents spurious utual coupling to external circuit eleents, e.g., currents in 19

30 conductors leading to or fro the device in question, and as such will induce a voltage that is coherent with the flux in the core. For ratio windings, an induced voltage is usually inconsequential if it is sall with respect to the noral voltage appearing across the burden (for secondary windings) or at the source (for priary windings). However, for the detection winding of a current coparator, the voltage represents a direct error, and for tertiary and copensation windings that operate at low voltages, it can create probles as well. The solution to the proble is to add a single return loop around the window that cancels the single turn of the winding, as illustrated in fig Figure 5.2 olution for ingle-turn roble 5.4 Turns Counting oewhat surprisingly, it isn t necessarily easy to ake an error-free count of the nuber of turns that are being anually applied to a core. Ten turns are easy to count, but 100 turns requires soe care, and 1000 turns is even ore difficult. After the turns are applied, it is iportant to verify the count or else the device ay be useless. The setup shown in fig. 5.3 is useful for this purpose. Figure 5.3 Turns Counting Circuit 20

31 n the figure, X represents the nuber of turns in the winding to be counted and C represents the nuber of turns, presued known, of another winding placed on the sae core. The winding under test is excited by a source, and the voltage across each winding is accurately easured. For the results to be accurate, C should be readily countable, but no saller than 10; the test frequency should be chosen such that 10, (5.1) x X X where x is the agnetizing ipedance of the winding under test at that frequency x is the leakage ipedance of the winding under test, and V 1 should be set high enough that V 2 is readily easurable with uncertainty less than 1/(10 X ). The nuber of turns in the test winding is then given by: V1 X round C, (5.2) V2 where round[*] represents the nearest integer value of *. Clearly, as X approaches 1000 or ore, the deands on the volteter s accuracy becoe critical. f a volteter with sufficient accuracy is not available, then it becoes necessary to create a bridge circuit in which the voltage ratio is copared with the prograable voltage ratio of an accurate, inductive voltage divider. 6. AVE AD ACTVE CORRECTO TECHQUE As we saw in earlier chapters, the ratio accuracy of a siple two-winding current transforer is liited by the inherent agnetizing ipedance of the secondary winding. The result is that a sall portion of the secondary current is shunted away fro the burden. A nuber of approaches have been proposed to iniize this error, and these are generally based on adding corrective cores and windings, or on the use of feedback aplifiers, or ore often, on cobinations of both approaches. 6.1 assive Two-tage Current Transforers The addition of a second core and winding as illustrated in fig. 6.1 creates a so-called two-stage transforer [10, 26]. As the figure illustrates, the priary and secondary windings link both cores, but the tertiary winding only links core 2. The second stage consisting of core 2 and the tertiary winding, senses the apere-turn difference of the first stage, i.e., the agnetizing current, and under the proper conditions produces a tertiary current that is very nearly equal to it. Therefore, the su of the secondary and tertiary winding currents is very nearly equal to the ideal secondary current. For the second stage to produce an accurate correction however, the two stages ust have separate burdens, or the coon burden ust be very sall. The equivalent circuit of fig. 6.2 illustrates the case in which there are two separate burdens, designated B1 and B2. ote that the equivalent circuit is siply the equivalent circuit of a siple current 21

32 Figure 6.1 Two-tage Current Transforer with eparate Burdens transforer (stage 2) ebedded in the equivalent circuit of another siple current transforer (stage 1), and the priary current of stage 2 is the agnetizing current of stage 1. A solution of the network equations for fig. 6.2 gives the following expression for the transforer ratio, defined as the ratio of the priary current to the su of the actual secondary and tertiary currents: 1 1 2, (6.1) ' ' T where B1 T B2 1 and 2 and ( B1) 1, ( T B2 ) 2. (6.2) 1 2 Figure 6.2 Two-tage Transforer with eparate Burdens: Equivalent Circuit Therefore, the ratio error as defined in section 1.2 is approxiately equal to inus the product of the ratio errors of the individual stages. Brooks and Holtz first described the two-stage current transforer [26] in n their application, two separate windings of a watteter constituted 22

33 the two burdens, and the watteter responded to the su of the two currents, thus providing a ore accurate reading than was attainable with a conventional watteter and single-stage transforer. Figure 6.3 Two-tage Transforer-caled Resistor Fig. 6.3 illustrates another application of the two-stage transforer with separate burdens that has been used successfully by the author and others. For the case in which B2 is approxiately equal to B1, this is a siple but very accurate way to produce a lower-valued trans-resistance fro a larger-valued 4-terinal resistor. Here, the output voltage is given by: V O B , (6.3) where 1 B2 B1 and 1 and 2 are as defined above. Although siple, this circuit can be quite accurate and exhibit wide bandwidth; furtherore, reasonable resistance values (0.1 to 1.0 oh) can be used for good ac accuracy. Figures 6.4a and 6.4b Two-tage Transforer with ingle Burden: Equivalent Circuits Also, the power dissipated in the output resistor is only p/s ties the power dissipated in an un-scaled resistor of the sae value. The accuracy of a two-stage transforer degrades usually 23

34 24 when a single burden is shared by both stages. The corresponding equivalent circuit is shown in figs. 6.4a and in a reduced for in 6.4b. Figure 6.4c Two-tage Transforer with ingle Burden: Final Equivalent Circuit Finally, following a Y transforation, the equivalent circuit in fig. 6.4b yields that shown in fig. 6.4c, With the usual assuption that the burden and leakage ipedances are sall with respect to the agnetizing ipedances, the circuit of fig. 6.4c yields the following transforer ratio: ' ' B B T B T. (6.4) ote that the effect of the second stage in this case is to essentially eliinate the contribution of the secondary winding leakage ipedance to the error, while leaving the contribution due to the burden untouched. Therefore, unless the burden ipedance is very sall, the two-stage transforer with coon burden is not in itself a very useful circuit. 6.2 Active Two-tage Transforers The use of feedback aplifiers can iniize the previously discussed errors in two-stage transforers caused by the burden ipedance. A siple exaple is shown in fig Here, the effective burden as seen by the transforer is reduced by the aplifier gain, G, and the output voltage is given by: 2 ' ' B B B T O G G G V. (6.5). and,, where T T T T T s