Inductances of coaxial helical conductors

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Computer Applications in Electrical Engineering Inductances of coaxial helical conductors Krzysztof Budnik, Wojciech Machczyński Poznań University of Technology 60-965 Poznań, ul.

1 Computer Applications in Electrical Engineering Inductances of coaxial helical conductors Krzysztof Budnik, Wojciech Machczyński Poznań University of Technology Poznań, ul. Piotrowo 3a, Wojciech Twisting of conductors is commonly used in various fields of electrical instruments and measurement systems in order to reduce the electromagnetic interference (EMI). Knowledge of inductances for helical conductors is needed for fundamental electromagnetic calculations, e.g. in electromagnetic compatibility studies. In the paper, a calculation method based on the Neumann s formula is applied for the inductance calculation of coaxial helical conductors of finite length. The exemplary calculations are also presented. 1. Introduction Helical conductors or coils have been commonly used in various electrical instruments and measurement systems, e.g. twisted bifilar leads, and twisted conductors, including superconducting composites and multistrand cables, Fig.l. Fig. 1. Twisted-wire pair and low-voltage power cable Generally, the effect of twisting can be utilized when the magnetic field is a problem. Twisting of conductors pairs is a well known method used in telephone communications in order to reduce the electromagnetic interference (EMI) and minimize crosstalk among lines in bundled cables. Already practiced is twisting of insulated high-voltage three phase power cables and single-phase distribution cables as well. Inductance calculations for helical conductors are needed for fundamental electromagnetic calculations. The inductance calculation of helical conductors has been studied analytically and numerically, using the integral expression for the vector potential of an infinitely long helical conductor [1-3]. In [4] the full analytical expression for the mutual inductance of long coaxial helical thin conductors have been studied using Neumann s formula for the whole range from 0 to w of the pitch length. In addition, an approximate expression for the selfinductance for a long helical round conductor has been derived using the summation of external and internal inductances. Few papers take up the problem of finite length helical conductors. The work [7] deals with numerical determination 358

2 of self and mutual inductances of double-helix coils. Parametric description of appropriate line integral has been used to determine the inductance by evaluating the vector potential integral along the edge of the twisted conductors. An attempt to evaluate the mutual inductance has failed however due to computational problems connected with multiple numerical integration. In [8, 9] the calculation inductances of finite length twisted conductors has been tackled. In the paper the mutual inductance formula of long coaxial helical conductors is presented and the calculation of the mutual inductance of finite length both righthanded and left-handed coaxial helical thin conductors, Fig.2, is studied using the Neumann s formula. A more general case of two coaxial helical conductors with different lengths, different winding radii, and different pitch distances is concerned. From the expression derived, the external inductance of a helical round conductor with finite length is calculated as approximately the mean of the mutual inductances of the filament at the inner or outer edge of the conductor and the central filament. In addition, an approximate expression for the self-inductance for a finite length helical round conductor is obtained using the summation of the external and internal inductances. Furthermore the inductive coupling between twisted wires in cables with grounded conductor is considered and the voltage induced by phase currents in the protective earth (PE) conductor is calculated. z z Fig. 2. Left-handed and right-handed helix 2. Mutual inductance of coaxial helical conductors Consider two helical conductors located coaxially in the Cartesian co-ordinate system (x, y, z), Fig.3. The mutual inductance between the helical current lines can be computed using the double integral Neumann s formula [11]: M = U dci dc2 (1) 4^ H r

3 Here, as shown in Fig. 3, c1and c2 are the contours of the filament structures, respectively, dci and dc2 are infinitesimally small integration elements, r is the distance between line elements dc1and dc2; the symbol /x0 denotes the magnetic permeability of the vacuum (4n-10-7 H/m) Infinitely long coaxial helical conductors An analytical expression for the mutual inductance of an infinitely long coaxial helical thin conductors have been studied using Neumann s formula for the whole range from 0 to w of the pitch length in [4] and [6]. The approximate formula for the mutual inductance of right-handed (Q = 2 n/h) and left-handed (Q = -2 n/h) thin helical conductors has been derived using with Neumann s formula. With the condition l >> a2> a1 the integration in (1) can be performed after some mathematical treatment analytically [5] and the result is obtained in the form of an infinite series. This analytical expression for the mutual inductance according to [6] takes the form: 360

4 M = Po- (ln -1) + Q 2 aj2+ 2% a2 4% LI l + f In ( n \ a1)kn (n Q a2) c o s [ n p - P 01)] + (2) % n=1 ^n+1 (n Q a 1) Kn+1 (n Q a 2) + c o s [ n p - p ^ ) ] 2% n=1 _+ In-1 (n a1) Kn-1 (n Q a 2) valid under the condition that h1 = h2 = h and l1 = l2 = l>> a2; In and Kn are the modified Bessel functions of the order n, Q = 2%/h Finite length coaxial helical conductors The more general case concerning two coaxial right-handed helical conductors with different lengths, different winding radii, and different pitch distances, as in Fig.3, can be tackled using numerical integration of (1). The calculation method will be explained in the following. Denoting the winding radius of the i-th (I = 1,2) helix by ai, the pitch distance of the helix by hi and the ę co-ordinate of the point where the helix intersects the plane z = 0 by ę 0i, the parametric equations of the i-th helical line with respect to the parameter ęi (ę0i < ęi < 2%li/hi+ę0l) and with ę m ^ 0 are: X, ( p ) = ai cos p Yi ( p ) = at sin p (3) h Z, (Pt ) = ~Z~ (P - P0i ), i = 1,2 2% In order to apply the formula (1), we have to find suitable expressions dci ( p ) and r(ęi). By looking at Fig.3, and taking into account the eqn.(3): dci ( p ) = -a i sin ptd p t 1x + at cos ptdpt 1 y + ^h - d p i 1z, 2% i = 1,2 (4) where 1 x,1 y,1 z are rectangular unit vectors. The scalar product when we are given the Cartesian components of the two vectors in ( 1) takes the form: where dc 1 dc 2 = dc1xdc 2x + dc^dc 2y + dc^dc 2 z (5) 361

5 dcix = -a i sin pidpi dciy = ai cos ptdpt (6) Remembering that dciz = ^ ~ d Pi, z 2n ' " i = 1, 2 r = { * 1f a ) - X 2(^2)]2 + f t p ) - Y2OP2)]2 + [ Z p - Z2(^2)]2}/2 (7) it follows from (3) that: 2 2 (a1cos (p1- a2 cos (p2) + (a1sin (p1- a2 sin (p2) + r = < + [h1(p1- P01) - h2(p2 - P02)]2 (8) [ 2n Finally, the mutual inductance between two coaxial helical conductors with different finite lengths l1 and l2, different pitch distances h 1 and h2 and different winding radii a1and a2, takes the form: 2n?1 2nl2 hi ho 1T +P01 ^ T +P02 [a 1a 2 cos(p1 - P2) ^]dp1d p 2 m =^ r r 4n J J (9) 2 2 p01 p02 a 1 + a2-2a1a2 cos(p1 - p2) + + [h1(p1 - P01) - h2(p2 - P02 ) ]2. [ 2n. The integral formula (9) has to be solved numerically. Consider next the case of two coaxial left-handed helical conductors with different lengths, different winding radii, and different pitch distances. Denoting the winding radius of the i-th (i=1,2) helix by ai, the pitch distance of the helix by hi and the ę co-ordinate of the point where the helix intersects the plane z=0 by ę ci, the parametric equations of the i-th helical line with respect to the parameter ęi (ęci<9i<2nli/hi+ęci) and with ę <#0 are: X i (p ) = ai cos Pi Yi (P,) = -a, sin p, h z, (P, ) = ~z~ (P, - P0iK i = 1,2 2n In order to apply the formula (1), we have taking into account the eqn.(10): h dci (pi) = -a i sin ptdpi 1x - ai cos ptdpi 1 y + ~ d p i 1z, i = 1,2 2n where 1 x,1 y,1 z are rectangular unit vectors. The scalar product when we are given the Cartesian components of the two vectors in ( 1) takes the form: 362 (10) (11)

6 where dc 1 dc 2 = dc1xdc 2x + dc1ydc 2y + dck dc2 z dcix = -a i sin ptdpi dciy = -a i cos pidpt (12) (13) Remembering that dciz = ^ ~ d Pl, 2% i = 1, 2 r = {[X 1 (p1) - X 2 (p2)]2 + [Y1(P1) - Y2 (p2)]2 + Z (P1) - Z 2 (14) it follows from (10) that: 2 2 (a1cos p1 - a2 cos p2) + (a2 sin p2 - a 1 sin p1) + r = < (15) + [h1(p1- P01) - h2(p2 - P02)]2 [ 2% Finally the mutual inductance between two left- handed coaxial helical conductors with different finite lengths l1and l2, different pitch distances h1and h2 and different winding radii a1and a2, takes the form: h +p01 h2 +p02 m = ^ r 4% p p01 \a1a 2 cos(p1 - p2) ^]dp1dp2 2 2 a 1 + a 2-2a 1a 2 cos(p1 - p2) + + \h1(p1- p 01) - h2(p 2 - p02)]2. [ 2%, Note that the integral formula (16) is identical as the formula (9). Consider next the system of right-handed and left-handed coaxial helical conductors with different lengths, different winding radii, and different pitch distances, as shown in Fig.4. Denoting the winding helix radii of by a 1and a2, the pitch distances by h1and h2, and the ę co-ordinate of the point where the helix intersects the plane z =0 by ę 01 and ę 02, of the right-handed and left-handed helical lines respectively, the parametric equations of the helical lines can be expressed as follows: X 1(p ) = a1cosp1 (16) Y1(p1) = a1sin p1 (17) for the right-handed, and Z1(p1) = ^L (p1- p01) 2% 363

7 X 2 (P2) = a 2 cos P2 Y2 (P2) = - a 2 sin P2 (18) Z2(P2) = }hr (P 2- P02) for the left-handed helical conductor. Fig. 4. Two coaxial helical current lines ct (right-handed) and c2 (left-handed) The scalar product in the formula (1) is given by: dc1 dc 2 = [-a 1a 2 cos(p1+ p2) + h h ]dp1d p 2 (19) 4n and the distance between line elements dc1and dc2: 2 2 (a1cos p1- a 2 cos p2) + (a1sin p1+ a 2 sin p2) + r = (20) + [h1(p1- P01) - h2(p2 - P02)]2 [ 2n Thus, the mutual inductance between the right-handed and left- handed coaxial helical conductors with different finite lengths l1 and l2, different pitch distances h1 and h2 and different winding radii a1and a2, is from (1), (19) and (20): 364

8 2nl1 2nl2 h, +Pt +p02, hh2- [-a1a2 cos(p1 + P2) ]dp1dp2 m = r r 4n2 4n P P 2 " p01 p02 a 1 + a 2-2a1a 2 cos(p1+ p2) + (21) + [h1(p1 - P01) - h2(p2 - P02)]2 [ 2n 3. Self-inductance of a helical conductor Consider next the calculation of the self-inductance of a single helical round conductor, Fig.5. One simple approximate method is to calculate the self-inductances as the sum of the external and internal inductances: L = Le + Li (22) 3.1. Infinitely long helical conductor Fig. 5. Finite length helical round conductor The Neumann s relation (1) cannot be used to determine the external inductance unless the finite radius of the helical wire is taken into account. Let us assume that the radius of the helical round conductor is r0. If the external inductance Le is approximated by the mean of two extreme mutual inductances of the inner helical filament located at (a-rc, ę, z = 0) and the central helical filament of the conductor passing through (a, ę, z = 0), and the outer helical filament passing through (a+rc, ę, z = 0) and the central helical filament is taken, the external inductance of a long, thin right-handed or left-handed helical conductor can be estimated, using (2) as follows [4], [6]: 365

9 2%,yja(a + r0) 8% + ^ f tn ( n (a - r0))kn (n\q a) + In (n\q a)kn (n\q (a + r()))}+ 2% n=1 + ^ Qq 2 a(a - r0) f n+1v 1 0" n+1v 1 1 ' 4% n=1_+ In-1 (n Q (a - r0)) K - (n Q a) (23) +» Ł Q 2 a(a + r0) f P "1 ^ K" * (n n (a + ^ + 4% n=1 + In-1(n Q a) Kn-1(n Q (a + r0)) The internal inductance Li calculated from the magnetic energy within the helical round conductor with length l, winding radius a and conductor radius r0 according to \4] can be obtained approximately using the formula: where the cross-sectional shape perpendicular to the central axis of the helical conductor is always assumed as circular with the radius r Finite length helical conductor In the general case of the helical conductor with finite length, the external inductance can be calculated as approximately the mean of the mutual inductances of the filament at the inner or outer edge of the conductor with the radius r0 and the central filament using the expression (9). In this case the equations (6) take the form: dc1x = -(a - ^ ) sin p d p dc2 x = -(a + r0)sinp2dp2 dc1y = (a - r0) cos p d p (24) dc2 y = (a + r0)cosp2dp2 (25) and the distances between line elements dc1and dc2 can be expressed as: 366

10 2 2 [a cos p1 - (a - r0) cos p 2 ] + [a sin p1 - (a - r0) sin p 2 ] + + [ h(p1 - P01 - P2 + P02) ]2 [ 2n. [acosp1- (a + r0)cosp 2]2 + [a sinp1 - (a + r0)sinp2]2 + r = + [ h(p1 - P01 - P2 + P02) ]2. [ 2n Thus, taking into account the relation (9): 2nn '1 hji'i-2 2 ni. fi 2 ~+Po1 +P02 [a(a - r0) cos(p1 - P2) + 2 ]dp1dp2 4 n L«1 = P I 2 2 p01 p02 a + (a - r0) - 2a(a - r0)cos(p - p 2) + (26) (27) 2nl1 2nl, h1 +P01 h2 +P02 L.2 = J e2 4n p p01 J p02 [ h(p1 - P01 - P2 + P02) ]2 [ 2n [a(a + r0) cos(p1 - P2) + 2 ]d P1d P2 4 n ' 2 2 a + (a + r0) - 2a(a + r0)cos(p - p 2) + 2 (28) + [ h(p1 - P01 - P2 + P02) ]2 [ 2n. and the external inductance of the helical conductor can be obtained as follows: Le = (Le1 + Le2)/2 (29) The self-inductance of the right-handed or left-handed helical conductors can be next calculated according to formulas (22), (24), (27), (28) and (29). It should be pointed out that in case h ^ 0, the helical line current degenerates to a conducting circular loop (conducting wire with the radius r0 bent into a circular loop of mean radius a), placed symmetrically in the xy plane with the center located at the origin 0, and curried the current I. Evaluation of the e.g. integral (27) shows that after some algebraic manipulations the external inductance of the circular loop can be written in the form: 2 2 Le = ^qva(a - r(j) (_ - k ) K - - E k k (30) where k 2 = 4a(a - ro) (2a - ro)2 (31) n/2 dp K = K (k) = J o ^1 - k 2 sin2 p (32) 367

11 is the complete elliptic integral of the first kind, and %/2 i E = E (k) = j V1 - k 2 sin2 p d p ( ) 0 ( ) is the complete elliptic integral of the second kind [12]. 4. Voltage induced in PE conductor by phase currents in twisted conductor cable with finite length Multi-core low voltage power cables consist of the phase conductors and of a neutral conductor and/or a protective earth (PE) conductor. The individual insulated conductors are twisted together and each conductor can be represented by a helical line. In twisted power cables with PE conductors a common mode voltage is induced by currents in the phase conductors. This is valid even in the case of balanced currents due to the fact that the PE conductor has a certain geometrical asymmetry with respect to the phase conductors. The voltage induced in the PE conductor can be calculated from the relation: U PE = ja (M Ll -Pe Ł 1 + M L2-PE1 2 + M L3-Pe Ł 3) (34) where h is a phasor current in the i-th phase, ML-PE represent a mutual inductance between a phase conductor and the PE conductor. The voltage induced into the PE-loop according to [10] is about 0.44 ^V/A/m/Hz, what was also verified by experimental investigations. This value, in the case of balanced three phase system, can be obtained from the simplified relationship for induced voltage UPE as follows: U PE = 2 %f I M NET (35) where MNET is the net mutual inductance derived from the superposition of the individual mutual inductances between phase conductor and the PE conductor ML-PE. It should be however pointed out, that the above definition is not precise: the net mutual inductance MNET should be calculated in the case of balanced currents as the modulus of the sum of complex numbers from the relationship: M NET = \M L1-PE + M L2-PE exp( - j 2%/3) + M L3-PE exp( j 2%/3) (36) The value of the voltage induced (0.44 ^V/A/m/Hz) gives for the frequency f = 50 Hz the net mutual inductance MNET = 70 nh per 1 A and per meter length and Upe = 22 ^V. 5. Model validation and calculation examples 5.1 Mutual inductance between two coaxial solenoids Mutual inductance per unit length between two coaxial solenoids of the same pitch lengths with winding radii of 10 and 12 mm and the azimuthal angular difference of 368

n/6 has been calculated in [1]. It follows from the calculations that the mutual inductance between the solenoids is about 100 ^H/m if the pitch length is 2 mm.

12 n/6 has been calculated in [1]. It follows from the calculations that the mutual inductance between the solenoids is about 100 ^H/m if the pitch length is 2 mm. The calculations according to the formula (9) have been curried out for the following parameters: a 1= 10 mm, a2 = 12 mm, h1= h2 = 2 mm, l1= l2 = 50 cm, ę 01 = 0, ę02 = 30, giving: M = 51.1 ^H. Since the mutual inductance is ^H/m. The agreement between the results is excellent Self-inductance of a helical conductor Self-inductance per unit length of a single helical conductor with the radius 0,245 mm, Q = 3.14 rad mm-1 and winding diameter 0.84 mm has been calculated in [4]. It follows from the calculations that the self-inductance of the conductor is about 1.75 ^H/m. The self-inductance of the helical conductor has been calculated according to formulas (22), (24), (27), (28) and (29). The calculations of the external inductance have been curried out for the following parameters: a 1= mm, a 2 = 0.42 mm, h1= h2 = 2 mm, l1= l2 = 1 m, ę01 = 0, ę02 = n, and a 1= 0.42 mm, a2 = mm, h1 = h2 = 2 mm, l1= l2 = 1 m, ę01 = 0, ę02 = n according to the formulas (27) and (28) respectively, giving: Le1 = 1.62 ^H and Le2 = 1.79 ^H. Since the external inductance is from (29) Le = ^H. The internal inductance of the conductor, according to (24), is Li = ^H. Thus the self-inductance of the conductor is L = 1.79 ^H. The agreement between the results is excellent Inductive coupling between wires in cables with grounded conductor The calculations of the mutual inductances ML-PE according to the formula (9) derived in the paper have been curried out for 1 m long twisted cable 4^25 mm2 and a 0.4 m pitch length, formed by four wires located 90 spatial degrees from each other as in [10]. The mutual inductances have been calculated numerically according to the formula (9) for following parameters: a 1= a2 = a3 = a4 = a = cm and 1.1 cm, h1= h2 = h3 = h4 = h = 40 cm, l1= l2 = l3 = l4 = 100 cm, ę01 = 0, ę02 = 90, ę03 = 180, ę04 = 270, and the voltages for I1= I2 = I3 = I = 1 A. The results of calculations are shown in the Table 1. It follows from the calculations that the voltage induced into the PE-loop at f =50 Hz and the net mutual inductance are in good agreement with the results obtained in [10]. It shall be however mentioned, that the simplified approach - relation (35) is valid in the case of balanced currents only. In other cases the use of the relation (35) can lead to improper results. Exemplary, in the case of slight asymmetry of phase currents: I1 = 1.03 A, I2 = 0.97 A, I3 = 1.0 A, the induced voltage in the PE conductor obtained from the relation (34) at f = 50 Hz, with the mutual inductances previously calculated, is UPE = (a = 5.65 mm) and UPE = 32 (a = 11 mm). The results are significantly away from the value 22 ^V. 369

13 Table 1. Mutual inductances between phase conductors and the PE conductor and the induced voltage in the PE conductor by balanced phase unit currents Helix radius Results according to (9), (36) and (34) Results according to [10] a = 5.65 mm Mli-pe=907 nh M L2-pe=836 nh M Ł3-pe=866 nh M net = 61.7 nh/a/m UPE = 19.4 ^V/A/m ( f =50 Hz) a = 11 mm Ml1-pe=H6 nh Ml2-pe=703 nh Ml3-pe=734 nh M net = 63.5 nh/a/m UPE = 19.9 ^V/A/m ( f =50 Hz) M net = 70 nh/a/m Upe = 22 ^V/A/m ( f =50 Hz) 5.4. Mutual inductance between coaxial right- and left-handed solenoids The use of the formulas obtained in the paper allows one to calculate the mutual inductances in more complex system of coaxial helical conductors. Exemplary consider the system of right-handed and left-handed coaxial helical solenoids with different lengths, different winding radii, and different pitch distances. The calculations of the mutual inductance have been curried out for the following parameters: a 1= 10 mm, h1= 2 mm, l1= 20 cm, ę01 = 0 (right-handed helix) and a2 = 12 mm, h2 = 4 mm, l2 = 25 cm, ę 02 = n/6 (left-handed helix), according to the formula (21), giving: M = 9.47 ^H. It should be mentioned that the approximate analytical and numerical methods [1-7] are not able to tackle such problems. 6. FINAL REMARKS Methods of the determination of the inductances both for infinitely long and finite length twisted bifilar lead are presented. The formulas based on the Neumann s relation for calculating the mutual inductance for finite length rightand left-handed helices with different lengths, different winding radii, and different pitch distances are derived. Verification of the method presented lies in comparison of calculation results with calculation results available in the literature. The formulas derived allow calculations of the mutual inductance of a two-wire helix, as well as the external inductance of single helical conductor with finite radius, and can be used by a software tool when the magnetic field is a problem, e.g. in electromagnetic compatibility studies when the inductive coupling between twisted wires in cables with grounded conductor is considered and the voltage induced by phase currents in the protective earth (PE) conductor has to be calculated. 370

14 References [1] Tominaka T., Inductance calculation for helical conductors, Superconductor Science and Technology, 18 (2005), pp [2] Tominaka T, Vector potential for a single helical current conductor, Nuclear Instruments andm ethods in Physics Research, A 523 (2004), pp [3] Tominaka T., Chiba Y., Low frequency inductance for a twisted bifilar lead, Journal o f Physics, D: Appl. Phys. 37 (2004), pp [4] Tominaka T., Self- and mutual inductances of long coaxial helical conductors, Superconductor Science and Technology, 21 (2008), pp [5] Tominaka T., Vector potential for a single helical current conductor, Nuclear Instruments and Methods in Physics Research, A 523 (2004), pp [6] Tominaka T., Current and field distributions of a superconducting power transmission cable composed of helical tape conductors, Superconductor Science and Technology, 22 (2009), pp [7] Franek J., Kollar M., Determination of self and mutual inductances of double-helix coil, Journal o f Electrical Engineering, vol. 60, no. 5, 2009, pp [8] Budnik K., Machczyński W., Calculation of inductances for twisted bifilar lead, XXXV International Conference on Fundamentals of Electrotechnics and Circuit Theory, IC-SPETO 2012, Gliwice-Ustroń, , pp [9] Budnik K., Machczyński W., Mutual inductance of finite length twisted-wire pair, Poznan University of Technology Academic Journals. Electrical Engineering, issue 69, Poznan 2012, pp [10] Jaekel B. W., Inductive coupling between wires in cables with grounded conductor, Progress in Electromagnetics Research Symposium PIERS, Moscow, Russia, August 18-21, 2009, PIERS Proceedings, pp [11] Griffiths D. J., Introduction to Electrodynamics, Prentice Hall, Englewood Cliffs, N.J., [12] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, New York: Denver,

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