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1 Michal Szype. "Inductance Measuement." Copyight 2000 CRC Pess LLC. <

2 Inductance Measuement Michal Szype Univesity of Mining and Metallugy 50.1 Definitions of Inductance 50.2 Equivalent Cicuits and Inductive Element Models 50.3 Measuement Methods Cuent-Voltage Methods Bidge Methods Diffeential Methods Resonance Methods 50.4 Instumentation Inductance is an electical paamete that chaacteizes electic cicuit elements (two- o fou-teminal netwoks) that become magnetic field souces when cuent flows though them. They ae called inductos, although inductance is not a unique popety of them. Electic cuent i (A) and magnetic flux Φ (Wb) ae intedependent; that is, they ae coupled. Inductance is a measuable paamete; theefoe, it has a physical dimension, a measuement unit (the heny), as well as efeence standads. Inductance is a popety of all electical conductos. It is found as self-inductance L of a single conducto and as mutual inductance M in the case of two o moe conductos. Inductos can vay in constuction, but windings in the fom of coils ae the most fequent. In this case, they have chaacteistic geometical dimensions: suface A, length l, and N numbe of tuns of windings. The path of magnetic flux can be specially shaped by magnetic coes. Figue 50.1 shows examples of diffeent inductive elements of electic cicuits. Figues 50.1a and b pesent windings with self-inductance made as a coeless coil (a), and wound on a feomagnetic coe that is the concentato of the magnetic field (b). A tansfome loaded by impedance Z L and an electomagnet loaded by impedance of eddy cuents in the metal boad, shown in Figues 50.1c and d, will have not only self-inductance of windings, but also mutual inductance between windings (c) o between winding and eddy cuents (d). Self-inductances and mutual inductances can be detected in busbas of electic powe stations as shown in Figue 50.1e, and also on tacks on pinted cicuit boads as in Figue 50.1f. Couplings between cuents and electomagnetic fields can be made delibeately but can also be hamful, e.g., due to enegy losses caused by eddy cuents, o due to electomagnetic distubances induced in the tacks of pinted cicuit boads. Inductos made as windings on feomagnetic coes have numeous othe applications. The pesence of a feomagnetic coe changes the shape of a magnetic field and inceases inductance; but in the case of closed coes, it also causes nonlinea elations between inductance and cuent as well as cuent fequency. Closed magnetic coes then have a nonlinea and ambiguous magnetization chaacteistic Φ (i) because of magnetic satuation and hysteesis. Inductances of open magnetic coes with an ai gap ae mainly dependent on the length of the magnetic path in the ai. Real coils made of metal (e.g., coppe) wie also show esistance R. Howeve, if esistance is measued at the coil teminals with ac cuent, it depends not only on the coss-section, length, and esistivity of the wie, but also on losses

3 FIGURE 50.1 Examples of diffeent inductive elements: coeless coil (a), coil with feomagnetic concentating coe (b), tansfome (c), electomagnet (d), element of electical powe station bus-bas (e), and pinted cicuit boad with conductive tacks (f). of active powe in the magnetic coe. These losses depend on both the cuent value and fequency. Resistances of inductive elements can also depend on the skin effect. This phenomenon consists of the flow of electic cuent though a laye of the conducto, nea its oute suface, as the esult of the effects of the conducto s own magnetic field geneated by the cuent flowing inside the conducto. Notice that the coils have also intetuns and stay (to Eath) capacitances C. Fom its teminals, the inductive elements can then be descibed by impedances (two-teminal netwoks) o tansmittances (fou-teminal netwoks), values that ae expeimentally evaluated by cuent and voltage measuements, o by compaison them with efeence impedances. The measued values and the equivalent cicuit models of inductive elements ae used fo evaluation of model paametes: self-inductances, mutual inductances, esistances, and capacitances Definitions of Inductance Self-inductance is defined as the elation between cuent i flowing though the coil and voltage v measued at its teminals [1]. v = L di dt (50.1) Using Equation 50.1, the unit of inductance, i.e., the heny, can be defined as follows: One heny (1 H) is the inductance of a cicuit in which an electomotive foce of one volt (1 V) is induced, when the cuent in the cicuit changes unifomly by one ampee (1 A) pe second (1 s). The definition of the unit implies the use of unifom-amp cuent excitation (the deivative is constant); in pactice, howeve, mainly sinusoidal cuent excitation is used in inductance measuement. Mutual inductance M of windings coupled by magnetic flux Φ is a paamete that depends on the coupling coefficient. The coupling coefficient is defined as pefect coupling in the case in which the total flux of one winding links the second one; patial coupling is the case in which only a faction of flux links the second one; and zeo coupling is the case in which no pat of the flux of one winding links the second one. Assuming that, as in Figue 50.1c, the second N s winding is not loaded (i.e., i s = 0) and the flux Φ is a pat of the total flux poduced by cuent i p, then voltage v s is descibed by:

4 FIGURE 50.2 Basic equivalent cicuits of inductive elements with self-inductance L, mutual inductance M fo low (LF) and high (HF) fequencies. dφ di v N k L L M i p d p s = s =± p s =± d t dt dt (50.2) whee ±k is the coupling coefficient of the pimay (p) and seconday (s) windings. Its sign depends on the diection of the windings. Because of the similaity between Equations 50.1 and 50.2, mutual inductance is defined similaly to self-inductance. The definitions of self-inductance and mutual inductance descibed above ae coect when L and M can be assumed constant, i.e., not depending on cuent, fequency, o time Equivalent Cicuits and Inductive Element Models Equivalent cicuits of inductive elements and thei mathematical models ae built on the basis of analysis of enegy pocesses in the elements. They ae as follows: enegy stoage in the pats of the cicuits epesented by lumped inductances (L) and capacitances (C), and dissipation of enegy in the pats of the cicuits epesented by lumped esistances (R). Essentially, the above-mentioned enegy pocesses ae neve lumped, so the equivalent cicuits and mathematical models of inductive elements only appoximate eality. Choosing an equivalent cicuit (model), one can influence the quality of the appoximation. In Figue 50.2, the basic models of typical coeless inductive elements of electic cicuits ae pesented, using the type of inductance (self-inductance L, mutual inductance M) and fequency band (low LF, high HF) as citeia. Models of inductive elements with feomagnetic coes and poblems concening model paamete evaluation ae beyond the scope of this chapte. The equivalent cicuits of coeless inductive elements at low fequencies contain only inductances and seies esistances. At high fequencies, paallel and coupling equivalent capacitances ae included. Calculations of LCR values of complicated equivalent cicuits (with many LCR elements) ae vey tedious, so often fo that pupose a special dedicated pocesso with appopiate softwae is povided in the measuing device. In metology, complex notation [1] is fequently used to descibe linea models of inductive elements. In complex notation, the impedances and tansmittances ae as follows:

5 2 2 X Z = Rz + jxz = Zm exp ( jφz), Zm = Rz + Xz, φz = actan R z z (50.3) 2 2 X T = RT+ jxt = Tm exp ( jφt), Tm = RT+ XT, φt = actan R T T (50.4) The components of impedances and tansmittances can be descibed as algebaic functions of the LCR elements detemined fo the equivalent cicuit. In each case, the foms of the functions depend on the assumed equivalent cicuit. By measuing the eal and imaginay components o modules and phase angles of impedances o tansmittances and compaing them to coesponding quantities detemined fo the equivalent cicuit, and then solving the obtained algebaic equations, the LCR values of the equivalent cicuit can be detemined. An example fo the LF equivalent cicuit in Figue 50.2 can be obtained: Z = R+ jωl= R + jx R= R, ωl= X z z z z (50.5) T = jωm = R + jx R = 0, ωm = X T T T T (50.6) Fo the HF equivalent cicuits shown in Figue 50.2, the models ae moe complex. Moe paticulaly, models of cicuits with mutual inductance can epesent: Ideal tansfomes, with only thei self-inductances and mutual inductances of the coils Pefect tansfomes, without losses in the coe Real tansfomes, having a specific inductance, esistance, and capacity of the coils, and also a cetain level of losses in the coe The quality facto Q and dissipation facto D ae defined fo the equivalent cicuits of inductive elements. In the case of a cicuit with inductance and esistance only, they can be defined as follows: 1 ω L Q = = = τω D R (50.7) whee paamete τ is a time constant. The pesented models with lumped inductance ae not always a sufficiently good appoximation of the eal popeties of electic cicuits. This paticulaly applies to cicuits made with geometically lage wies (i.e., of a significant length, suface aea, o volume). In such cases, the models applied use an adequately distibuted inductance (linealy, ove the suface, o thoughout the volume). Inductances detemined to be the coefficients of such models depend on the geometical dimensions and, in the case of suface conductivity o conduction by the suface o volume, by the fequencies of the cuents flowing in the conducto lines. A complex fomulation is used to epesent and analyze cicuits with lumped and linea inductances. Sometimes the same thing can be done using simple linea diffeential equations o the coesponding integal opeatos. The analytical methodology fo such cicuits is descibed in [1]. In the case of nonlinea inductances, the most fequently used method is the lineaized equivalent cicuit, also epesented in a simplified fom using a complex fomulation. Cicuits with distibuted inductance ae epesented by patial diffeential equations.

6 FIGURE 50.3 Cicuit diagam fo impedance measuement by cuent and voltage method Measuement Methods Impedance (o tansmittance) measuement methods fo inductos ae divided into thee basic goups: 1. Cuent and voltage methods based on impedance/tansmittance detemination. 2. Bidge and diffeential methods based on compaison of the voltages and cuents of the measued and efeence impedances until a state of balance is eached. 3. Resonance methods based on physical connection of the measued inducto and a capacito to ceate a esonant system. Cuent Voltage Methods Cuent voltage measuement methods ae used fo all types of inductos. A cuent voltage method using vecto voltmetes is shown in Figue It is based on evaluation of the modules and phase angles of impedances (in the case of self-inductance) o tansmittances (in the case of mutual inductance) using Equations 50.8 and 50.9 Z v R v R V ( j ) R V 2 2 m2exp φ2 = = = j Z j (50.8) i v V ( j ) = m2 ( V exp φ2 φ 1 )= m exp( φz) 1 m1exp φ1 m1 T v R v R V ( j ) R V s s ms exp φs = = = j T j (50.9) i v V ( j ) = ms ( V exp φs φ p)= m exp( φt) p p mp exp φp mp whee R = Sampling esisto used fo measuement of cuent v 1 = Voltage popotional to the cuent v 2 = Voltage acoss the measued impedance v p = Voltage popotional to pimay cuent v s = Voltage of the seconday winding of the cicuit with mutual inductance A block diagam illustating the pinciple of the vecto voltmete is shown in Figue 50.4a. The system consists of the multiplie o gated synchonous phase-sensitive detecto (PSD) [3, 4] of the measued

7 FIGURE 50.4 Block diagam (a) and phaso diagam (b) illustating the pinciple of opeation of a vecto voltmete. Block abbeviations: Det phase-sensitive amplitude detecto, Int integato, Volt voltmete, Poc pocesso, Ph. mult contolled phase multiplexe. voltage v with the switching system of the phase of the efeence voltage v dn, integato, digital voltmete, and pocesso. The pinciple of vecto voltmete opeation is based on detemination of the magnitude V m and phase angle φ of the measued voltage v in efeence to voltage v 1, which is popotional to the cuent i. Assume that voltages v and v dn ae in the following foms: v = V sin ( ωt + φ )= V sin ωt cos φ+ cos ωt sin φ (50.10) m m( ) vdn = V m d t+ n π sin ω, n= 0123,,, (50.11) 2 Phase angle nπ/2 of voltage v dn can take values fom the set {0, π/2, π, 3/2 π} by choosing the numbe n that gives the possibility of detecting the phase angle φ in all fou quadants of the Catesian coodinate system, as is shown in Figue 50.4b. A multiplying synchonous phase detecto multiplies voltages v and v dn and bilinealy, and the integato aveages the multiplication esult duing time T i. V in Ti 1 = vv T i 0 dn d t (50.12) Aveaging time T i = k T, k = 1,2,3 is a multiple of the peiod T of the measued voltage. Fom Equations though 50.12, an example fo 0 φ π/2 (e.g., n = 0 and n = 1), a pai of numbes is obtained: V = 05. V V cos φ, V = 05. V V sin φ (50.13) i 0 m md i 1 m md which ae the values of the Catesian coodinates of the measued voltage v. The module and phase angle of voltage ae calculated fom: V m Vi 1 = Vi 0 + Vi 1, φ = actan V V md i 0 (50.14) Both coodinates of the measued voltage v can be calculated in a simila way in the emaining quadants of the Catesian coodinate system. A vecto voltmete detemines the measued voltage as vecto (phaso) by measuement of its magnitude and angle as shown in Figue 50.4.

8 FIGURE 50.5 Block diagam (a) and phaso diagam (b) of the thee-voltmete method. Opeational and diffeential amplifies ae epesented by blocks OA and DA. The cuent and voltage method of impedance o tansmittance measuement is based on measuement of voltages v 1 = i R and v 2 o v p = i R and v s, and the use of Equations 50.8 o Calculation of the voltage measuement esults and contol of numbe n is pefomed by a pocesso. Refeences [11 13] contain examples of PSD and vecto voltmete applications. Eos of module and phase angle measuement of impedance when using vecto voltmetes ae descibed in [11] as being within the ange of 1% to 10% and between 10 5 ad and 10 3 ad, espectively, fo a fequency equal to 1.8 GHz. Publications [9] and [10] contain desciptions of applications of compaative methods which have had an impotant influence on the development of the methods. Anothe method of compaative cuent/voltage type is a modenized vesion of the thee-voltmete method [7]. A diagam of a measuement system illustating the pinciple of the method is shown in Figue 50.5a and b. The method is based on the popeties of an opeational amplifie (OA), in which output voltage v 2 is popotional to input voltage v 1 and to the atio of the efeence esistance R to measued impedance Z. The phaso diffeence v 3 of voltages v 1 and v 2 can be obtained using the diffeential amplifie (DA). The thee voltages (as in Figue 50.5b) can be used fo the module Z m and phase ϕ calculation using elations: v R V1 = ir = Z = Z V R υ m (50.15) 2 2 V1 + V2 V v3 = v1 v2 φ= accos 2VV (50.16) whee V 1, V 2, V 3 ae the esults of ms voltage measuements in the cicuit. The advantage of the method lies in limiting the influence of stay capacitances as a esult of attaching one of the teminals of the measued impedance to a point of vitual gound. Howeve, to obtain small measuement eos, especially at high fequencies, amplifies with vey good dynamic popeties must be used. Joint eos of inductance measuement, obtained by cuent and voltage methods, depend on the following factos: voltmete eos, eos in calculating esistance R, system factos (esidual and leakage inductances, esistances and capacitances), and the quality of appoximation of the measued impedances by the equivalent cicuit.

9 FIGURE 50.6 Bidge cicuits used fo inductance measuements: Maxwell-Wien bidge (a), Hay bidge (b), Caey- Foste bidge (c), and ac bidge with Wagne banch (d). Block abbeviations: Osc oscillato, Amp amplifie, Det amplitude detecto. Bidge Methods Thee ae a vaiety of bidge methods fo measuing inductances. Bidge pinciples of opeation and thei cicuit diagams ae descibed in [2 5] and [7]. The most common ac bidges fo inductance measuements and the fomulae fo calculating measuement esults ae shown in Figue The pocedue efeed to as bidge balancing is based on a pope selection of the efeence values of the bidge so as to educe the diffeential voltage to zeo (as efeed to the output signal of the balance indicato). It can be done manually o automatically.

10 The condition of the balanced bidge v 0 = 0 leads to the following elation between the impedances of the bidge banches; one of them (e.g., Z l ) is the measued impedance: Z Z = Z Z ( R + jx )( R + jx )=( R + jx ) R + jx ( ) (50.17) Putting Equation into complex fom and using expessions fo the impedances of each banch, two algebaic equations ae obtained by compaing the eal and imaginay components. They ae used to detemine the values of the equivalent cicuit elements of the measued impedance. In the most simple case, they ae L and R elements connected in seies. Moe complicated equivalent cicuits need moe equations to detemine the equivalent cicuit paametes. Additional equations can be obtained fom measuements made at diffeent fequencies. In self-balancing bidges, vecto voltmetes peceded by an amplifie of the out-of-balance voltage v 0 ae used as zeo detectos. The detecto output is coupled with vaiable standad bidge components. The Maxwell-Wien bidge shown in Figue 50.6a is one of the most popula ac bidges. Its ange of measuement values is lage and the elative eo of measuement is about 0.1% of the measued value. It is used in the wide-fequency band 20 Hz to 1 MHz. The bidge is balanced by vaying the R 2 and R 3 esistos o by vaying R 3 and capacito C 3. Some difficulties can be expected when balancing a bidge with inductos with high time constants. The Hay bidge pesented in Figue 50.6b is also used fo measuement of inductos, paticulaly those with high time constants. The balance conditions of the bidge depend on the fequency value, so the fequency should be kept constant duing the measuements, and the bidge supply voltage should be fee fom highe hamonic distotions. The dependence of bidge balance conditions on fequency also limits the measuement anges. The bidge is balanced by vaying R 3 and R 4 esistos and by switching capacito C 3. Mutual inductance M of two windings with self-inductances L p and L s can be detemined by Maxwell- Wien o Hay bidges. Fo this, two inductance measuements have to be made fo two possible combinations of the seies connection of both coupled windings: one of them fo the coesponding diections of the windings, and one fo the opposite diections. Two values of inductances L 1 and L 2 ae obtained as the esult of the measuements: L = L + L + 2 M, L = L + L 2 M 1 p s 2 p s (50.18) Mutual inductance is calculated fom: ( ) M = 025. L 1 L 2 (50.19) The Caey-Foste bidge descibed in Figue 50.6c is used fo mutual inductance measuement. The self-inductances of the pimay and seconday windings can be detemined by two consecutive measuements. The expessions pesented in Figue 50.6c yield the magnetic coupling coefficient k. The bidge can be used in a wide fequency ange. The bidge can be balanced by vaying R 1 and R 4 esistances and switching the emaining elements. Fo coect measuements when using ac bidges, it is essential to minimize the influence of hamful couplings among the bidge elements and connection wies, between each othe and the envionment. Elimination of paallel (common) and seies (nomal) voltage distotions is necessay fo high measuement esolution. These couplings ae poduced by the capacitances, inductances, and paasitic esistances of bidge elements to the envionment and among themselves. Because of thei appeaance, they ae called stay couplings. Seies voltage distotions ae induced in the bidge cicuit by vaying common electomagnetic fields. Paallel voltage distotions ae caused by potential diffeences between the efeence point of the supply voltage and the points of the out-of-balance voltage detecto system.

11 Magnetic shields applied to connection wies and bidge-balancing elements ae the basic means of minimizing the influence of paasitic couplings and voltage distotions [8]. All the shields should be connected as a sta connection; that is, at one point, and connected to one efeence ( gound ) point of the system. Fo these easons, amplifies of out-of-balance voltage with symmetic inputs ae fequently used in ac bidges, as they eject paallel voltage distotions well. When each of the fou nodes of the bidge has diffeent stay impedances to the efeence gound, an additional cicuit called a Wagne banch is used (see Figue 50.6d). By vaying impedances Z 5 and Z 6 in the Wagne banch, voltage v c can be educed to zeo; by vaying the othe impedances, voltage v 0 can also be educed to zeo. In this way, the bidge becomes symmetical in elation to the efeence gound point and the influence of the stay impedances is minimized. The joint eo of the inductance measuement esults (when using bidge methods) depends on the following factos: the accuacy of the standads used as the bidge elements, mainly standad esistos and capacitos; eos of detemining the fequency of the bidge supplying voltage (if it appeas in the expessions fo the measued values); eos of the esolution of the zeo detection systems (eos of state of balance); eos caused by the influence of esidual and stay inductances; esistances and capacitances of the bidge elements and wiing; and the quality of appoximation of the measued impedances in the equivalent cicuit. The eos of equivalent esistance measuements of inductive elements using bidge methods ae highe than the eos of inductance measuements. The numbe of vaious existing ac bidge systems is vey high. Often the bidge system is built as a univesal system that can be configued fo diffeent applications by switching elements. One of the designs of such a system is descibed in [3]. Refeence [13] descibes the design of an automatic bidge that contains, as a balancing element, a multiplying digital-to-analog convete (DAC), contolled by a micocontolle. Diffeential Methods Diffeential methods [7] can be used to build fully automatic digital impedance metes (of eal and imaginay components o module and phase components) that can also measue inductive impedances. Diffeential methods of measuement ae chaacteized by small eos, high esolution, and a wide fequency band, and often utilize pecise digital contol and digital pocessing of the measuement esults. The pinciple of diffeential methods is pesented though the example of a measuing system with an inductive voltage divide and a magnetic cuent compaato (Figue 50.7). An inductive voltage divide (IVD) is a pecise voltage tansfome with seveal seconday winding taps that can be used to vay the seconday voltage in pecisely known steps [7]. By combining seveal IVDs in paallel, it is possible to obtain a pecise voltage division, usually in the decade system. The pimay winding is supplied fom a sinusoidal voltage souce. A magnetic cuent compaato (MCC) is a pecise diffeential tansfome with two pimay windings and a single seconday winding. The pimay windings ae connected in a diffeential way; that is, the magnetic fluxes poduced by the cuents in these windings subtact. The output voltage of the seconday winding depends on the cuent diffeence in the pimay windings. MCCs ae chaacteized by vey high esolution and low eo but ae expensive. In systems in common use, pecise contol of voltages (IVD) is povided by digitally contolled (sign and values) digital voltage divides (DVD), and the magnetic compaato is eplaced by a diffeential amplifie. The algoithms that enable calculation of L and R element values of the seies equivalent cicuit of an inductive impedance Z in a digital pocesso esult fom the mathematical model descibed in Figue When the system is in a state of equilibium, the following elations occu: v0 = 0 iz i = 0 (50.20) 1 b iz = v1, i = D v1 aj C (50.21) R + j L ω ω R

12 FIGURE 50.7 Scheme of the diffeential method. Block abbeviations: Osc, Amp as in Figue 50.6, Vect. voltm vecto voltmete, Poc pocesso, DVD digital voltage divide. whee 0 < D 1 is the coefficient of v 1 voltage division and the values a and b ae equivalent to the binay signals used by the pocesso to contol DVD. Multiple digital-to-analog convetes (DACs) ae used as digitally contolled voltage divides. They multiply voltage D v 1 by negative numbes ( a), which is needed in the case of using a standad capacito C fo measuements of inductance. Afte substituting Equation into Equation and equating the eal and imaginay pats, the following fomulae ae obtained: 2 ar C L = 2 2 D b + a ω 2 R 2 C ( ) 2 br R = D b + a ω R C ( ) (50.22) (50.23) The lengths N of the code wods {a n } a and {b n } b, n = 1, 2,, N detemine the ange and esolution of the measuement system; that is, the highest measuable inductance and esistance values and the lowest detectable values. The ange can be chosen automatically by changing the division coefficients D. Achieved accuacy is bette than 0.1% in a vey lage ange of impedances and in a sufficiently lage fequency ange. Measuements can be peiodically epeated and thei esults can be stoed and pocessed. Resonance Methods Resonance methods ae a thid type of inductance measuement method. They ae based on application of a seies o paallel esonance LC cicuits as elements of eithe a bidge cicuit o a two-pot (fouteminal) T -type netwok. Examples of both cicuit applications ae pesented in Figue In the bidge cicuit shown in Figue 50.8a, which contains a seies esonance cicuit, the esonance state is obtained by vaying the capacito C, and then the bidge is balanced (v 0 = 0) using the bidge esistos. Fom the esonance and balance conditions, the following expessions ae obtained:

13 FIGURE 50.8 Cicuits diagams applied in esonance methods: bidge cicuit with seies esonance cicuit (a), and two-pot shunted T type with paallel esonance cicuit (b). Block abbeviations as in Figue L = 1, R R R C = 2 2 ω R 4 3 (50.24) To calculate the values of the LR elements of the seies equivalent cicuit of the measued impedance, it is necessay to measue (o know) the angula fequency ω of the supply voltage. The fequency band is limited by the influence of unknown intetun capacitance value. In the shunted T netwok pesented in Figue 50.8b, the state of balance (i.e., the minimal voltage v 0 value) is achieved by tuning the multiloop LCR cicuit to paallel esonance. The cicuit analysis [7] is based on the sta-delta tansfomation of the C R C element loop and leads to the elations fo L and R values: L = 2, R = ω C ω C R (50.25) Accoding to efeence [7], a double T netwok can be used fo inductive impedance measuements at high fequencies (up to 100 MHz) Instumentation Instuments commonly used fo inductance measuements ae built as univesal and multifunctional devices. They enable automatic (tiggeed o cyclic) measuements of othe paametes of the inductive elements: capacitance, esistance, quality, and dissipation factos. Equipped with intefaces, they can also wok in digital measuing systems. Table 50.1 contains a eview of the basic featues of the instuments called LCR metes, specified on the basis of data accessible fom the Intenet [15]. Pope design of LCR metes limits the influence of the factos causing eos of measuement (i.e., stay couplings and distotions). The esults of inductance measuements also depend on how the measuements ae pefomed, including how the measued inducto is connected to the mete. Cae should be taken to limit such influences as inductive and capacitive couplings of the inductive element to the envionment, souces of distotion, and the choice of opeating fequency.

14 TABLE 50.1 Basic Featues of Selected Types of LCR Metes Manufactue, model (designation) Measuement ange of inductance Fequency Basic accuacy Pice Leade LCR µh 1100 H int. 1 khz 0.5% $545 (LCR bidge) ext. 50 Hz 40 khz Electo Scientific 200 µh 200 H 1 khz (3.5 digit) $995 Industies 253 (Digital impedance mete) Stanfod RS 0.1 nh 100 kh 100 Hz 10 khz 0.2% $1295 SR 715 ($1425) (LCR mete) Wayne Ke nh 10 kh 120 Hz 100 khz 0.1% $3500 Geneal Radio mh H 12 Hz 100 khz 0.02% $4120 (Pecision LCR mete) Hewlett-Packad 4284A 0.01 nh kh 20 Hz 1 MHz 0.05% $9500 (Pecision LCR mete) Refeences 1. R. C. Dof, Intoduction to Electic Cicuits, New Yok: John Wiley & Sons, J. P. Bentley, Pinciples of Measuement Systems, 2nd ed., Halow: Longman Goup U.K., New Yok: John Wiley & Sons, A. D. Helfick and W. D. Coope, Moden Electonic Instumentation and Measuement Techniques, Englewood Cliffs, NJ: Pentice-Hall, L. D. Jones and A. F. Chin, Electonic Instuments and Measuements, 2nd ed., Englewood Cliffs, NJ: Pentice-Hall, M. U. Reissland, Electical Measuement, New Yok: John Wiley & Sons, P. H. Sydenham (ed.), Handbook of Measuement Science, Vol. 1, Theoetical Fundamentals, New Yok: John Wiley & Sons, P. H. Sydenham (ed.), Handbook of Measuement Science, Vol. 2, Pactical Fundamentals, New Yok: John Wiley & Sons, R. L. Bonebeak, Pactical Techniques of Electonic Cicuit Design, 2nd ed., New Yok: John Wiley & Sons, S. Hashimoto and T. Tamamua, An automatic wide-ange digital LCR mete, Hewlett-Packad J., 8, 9 15, T. Wakasugi, T. Kyo, and T. Tamamua, New multi-fequency LCZ metes offe highe-speed impedance measuements, Hewlett-Packad J., 7, 32 38, T. Yonekua, High-fequency impedance analyze, Hewlett-Packad J., 5, 67 74, M. A. Atmanand, V. J. Kuma, and V. G. K. Muti, A novel method of measuement of L and C, IEEE Tans. Instum. Meas., 4, , M. A. Atmanand, V. J. Kuma, and V. G. K. Muti, A micocontolle- based quasi-balanced bidge fo the measuement of L, C and R, IEEE Tans. Instum. Meas., 3, , J. Gajda and M. Szype, Electomagnetic tansient states in the mico esistance measuements, Systems Analysis Modeling Simulation, Amstedam: Oveseas Publishes Association, 22, 47 52, Infomation fom Intenet on LCR metes.

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