surface current experiment

00 0 Contract 100014.17.A.020.-0015 HI-371-018 Microwave thysics Braoch Ballistic Research Lahoratorlt," U.S. Army Aberdeen Proving Ground PROXIMITY EFFECTS FOR PARALLEL RECTANGULAR CONDUCTORS IN NON-TRAIISMISSION-LINE MODE ......... ,g, By Iokert We Berton February 1972 r' r-rr)Fnn ,nhrm-P * JUN 21 1972Technical Report No. 620 I*~L L I5Th C This document has been approved for pablic release and s&le, its distribution is unlirnited. Division of Engineering and Applied PhysicsHarvard University -Cambridge, massachusetts Reproduced by NATIONAL TECHNICAL INFORMATION SERVICE U S Deparlment of Coemmetce Spingfeld VA 22151 SUnclas Csified ? DOCUMENT CON ZROL DATA .R & D $Security clas~licatl ,.of of title. bady of abstract and indexing annototion mumr be entoe-rd when tMe verall repor s% ri.en , # ORIGINA TIN(C ACVIvrv (Cetpero auJthore) Is. RrtEPORT SCUROTYv CLASSIFI CaTIONDivision of Engineering and Applied Physics Harvard University Cambridge, Massachusetts 02138 3 REPORT TITLE PROXIMITY EFFECTS FOR PARALLEL RECTANGULAR CONDUCTORS IN NON-TR NSMISSION-LINE MODE 4 DESCRIPTiVE NOTIES (7ype of ?epaet and.inClusive datea) Interim technical report S AU THONIS) (irite# name, middle initial, late name) -.Robert W. -xton 6-Wo,APOn ,,. 7. TOTAL NO OF PAGS ,. No. or tvs February 1972 37J 6S I -"O4TRqACT OR GRANT NO. IM.ORIGINATOR*IS REPiORT NUI4111[;NS) N00014-67-A-0298-0005 I. P, ojItcy NO. c. ob. oTNEO %REPOft'r NOISI (Any other numbers that may be assogned this report) d. " 10 OISIGUTION STATEMENT. This document has been approved for public release and sale; its distribution is unlimited. II SUPPL..MCNTARY NOTES.eoisyrA k1IAIY SiTIcs Pf~ ~ ~ '" Joi~nt e'rvices -tle~ctromcs P'rogramI through (Adm. Service -Office of Naval ___ __ ___ __ __Research, Air Force Office of Scientific 13 ASTRACT Bee- n. 1I S A-r¶.y E1et_ Command .- In applications of electrically short antennas such as multiturn loops, variation in ohmic losses due to proximity effects can significantly affect un- corrected estimates of efficiency. Thii paper investigates experimentally thei proximity effects on the current distributions and associated ohmic losses for two ~ identical parallel rectangular cross-section conductors as a function of separation distance and cross-section. The system is operated in a non-transmission-line mode. In the light of experimental results a first-order theory is developed and verified for analyzing relative proximity effects at the extremes of conductor separation distances. DD -"807" Unclassified\ 01 01-*807-6811 SecUrily Ctassificallon A-. 40 -~ -- ~* ., L Unclassified " " ttt Clsstt~lt't O ce ' "toLIINK A LlNKB L![INAC "I~~ ~ - OOI,/ RNOLCt OL WY Itow wY Proximity effects Parallel rectangular conductors Non-transmie sion-line mode Experiment and first-order theoryF II - - JI)D i~~ot,".,1473 (BACK) Unclassified S/ti ~OISO.AS2,Security claasificstion ~.................................................................- ! I 4 Office of Naval Research Contract N00014-67-A-0Z98-0005 NR-371-016 Microwave Physics Branch Ballistic Research Laboratories U. S. Army Aberdeen Proving Ground PROXIMITY EFFECTS FOR PARALLEL RECTANGULAR CONDUCTORS IN NON-TRANSMISSION.-LINE MODE By 2 Robert W. Burton Technicl Report No. 629 This document has been approved for public release and sale; its distribution is unlimited.1 February 1972 The research reported in this document was made possible throughsupport extended the Di\,sion of Engineering and Applied Physics, Harvard University by the U. S. Army Research Office, the U. S.Air Force Office of Scientific Research and the U. S. Office of Naval Research -under the Joint Services Electronics Program by Contracts N00014-67-A-0298-0006, 0005, and 0008. Division of Engineering and Applied Physics Harvard University Cambridge, Massachusetts I II1i1L d PROXIMITY EFFECTS FOR PARALLEI, RECTANGULAR , CONDUCTORS IN NON- TRANSMISSION- LINE MODE By Robert W. Burton 1 Division of Engineering and Applied Physics Harvard University Cambridge, Massachusetts ABSTRACT In applications of electrically short antennas such as multiturn loops, i variation in ohmic losses due to proximity effects can significantly affect uncorrected estimates of efficiency. This paper investigates experiment- ally the proximity effects on the current distributions and associated ohmic losses for two identical parallel rectangular cross-section conductors as a function of separation distance and cross-section. The system is operated in a non-transmission-line mode. * In the light of experimental results a first-order theory is developed and verified for analyzing relative proximity effects at the extremes of conductor separation distances. * *The author is with the Department of Electrical Engineering, U. S. Air Force Academy, Colorado. am -- I. Introduction A strong interest exists in determining the radiation efficiency of electrically small or minute antennas. In such cases the ratio of radiated power to ohmic losses is a significant parameter affecting calculations of efficiency. The efficiency of power transfer is given by 1.ff. = I + LOPr /P where L0 system ohmic los.,es Pr radiated power In* general ohmic losses are a function of the operating frequency, the macroscopic properties and geometry of the conducting material, and proximity effects due to the p:fesence of other current-carrying elements. For a given system, ohmic losses are typically calcua'ated by integrating the square of the current distribution of an isolated (.onciuctor and multiplying it by an appropriate equivalent resistance. More often than not a uniformly distributed current is assumed, or better a theoretically calculated or experimentally measured current distribution for an isolated conductor is employed. In electrically small multiturn loop antennas, however, proximity effects may dramatically alter the transverse current distribution of the isolated conductors. Recently Smith(1) showed theoretically that the errors in calculating the total power radiated by an electrically small two-turn loop made from a circular cross-section conductor can be as lar-e as 30% if proximity effects are neglected. I! For rectangular conductors ýFig. 1) the determination of proximicy effects is far from straightforward. However, some insight into the (Z) problem is given by Cockcroft .In this paper Cockcroft reviews the approach as auggested by Strutt( ) that the rectangular conductor be approximated by an ellipse of large eccentricity wherein at high frequencies (4)as pointed out by Kelvin the problem of the distribution of the fields becomes analogous to the electrostatic problem. Cockcroft applies this analogy directly to the rectangular conductor usin3 the Schwartz- Christoffel transformation. Briefly stated, Cockcroft slhowed that the a. c. resistance per unit length, R, in ohms per meter of an isolated rectangular conductor is given by R f 2(B/A) /-i- 2S8iAB where p ir the resistivity in ohm-meters, w the angular frequency; ý1 = 'rlio the absolute permeability with n0 -t 4 x 10-7 henry/mi; A and B the dimensions of the conductor; and f(B/A), a shape factor involving (5)elliptical integrals. More recently King examined the current distribution and impedance per unit length of a very thin isolated strip. The case for analyzing proximity effects between two parallel conductors, hfowever, involves intractable integrals and an exact analytical solution cannot be -found although numerical evaluation might be considered a useful substitute. Cockcroft's formula hae been converted to MKS units. I —| -4- In this paper a series cf experimental measuremcnts of surface current distributions are taken over two identical parallel rectangular conductors carrying equal currerts, as a function of separation distance and cross section. Relative ohmic losses due to proximity effects over the range of measuremenrs are calculated and gen~eralized to a significant set of cross-section geometries anad seIa ration distances. From these observa' ons a first-order theory, which extends the experimental data to inc'ude very small separation distances, is formulated and compared with the analytical solution developed by Cockcroft. II. Experimental Apparatus and Error Analysis The basic experimental apparatus (Fig. 2) employs a 100 KHz CW oscillator with an external shielded loop prosing system. The rectangular conductors are of sufficient length to be considered infinitely long in the sense that field contributions from current-carrying feed cables were kept less than 5% of the average probe reading from opposite sides. The particular geometry of a rectangular conductor does not lend itself to the con-Ventional approach of field probing using internally mounted probes with the consequence that an externally mounted shielded loop probe system (Fig. 3) was deuigned and raked across each conductlng face. The probe is constructed of 0. 021 inch 0. D. coaxial cable formed into a 1/16 inch by 1-1/2 inch rectangle. A uniform probe separation was accomplished using a 1/32 inch radius teflon bearing. The measure of isolation between the conductors and the probe system was derived by raking the probe first in one direction P'ad then the other (Fig. 4;. Excellent agreement exists except where the probe penetrates the high C . .'C ~ ~ CC.--C. C -- - _. Probe Oscillator 13 f eet Detector EXPERIMENTAL APPARATUS FIG..2 EXTERNAL PROBE FOR MEASURING SURFACE CURRENT FIG. 3,°°,°* °,*411 Ii I xc "D-Zx LL ILl . D KEY: I-7. A---t LB Sprobe as in A x probe as in B 0 1 2 DISTANCE S ALONG CONDUCTOR (INCHES) MEASURE OF PROBE ISOLATION FIG. 4 field at the edges of the conductor. In this area contributions from the current on the adjacent conductor face asymmetrically distort the probe readings. The overall error frorn this distortion generates an error of less than 3% in the calculations. In practice a rectangular loop probe senses field contributions derived from currents (I ) sampled over an area larger than that directly 5 below the probe. The area of sampled current as a function of probe height, geometry and distance from probe centerline is given by Is(X, ha 1) I i)Ix) dx dy (3)f fly + 01 with y, hl, h., x defined as in Figure 5. Assuming I(x) varies at most linearly over the sampled range, fifty percent of the probe (Fig. 3) readings are derived from currents up to 1/16 inch away from the probe centerline; and 85% from currents within 1/4 inch of the centerline(6) The results of this integration effect in the measurement of surface currents is most -,ignificant as the probe approaches the edge. If there were no contribution from he adjoining side, the actual current at the edge would simply be twice the measured value. In such a case with a uniform circumferential current distribution, the correction factor by which the measured values must be multiplied to give the actual surface current would be Correction Factor  2 -_ tn-(s/h) (4) 7T Y x2 '7 I zI PROBE GEOMETRY FIG. 5 \A - ''C~ 1 1 10 as derived from (3) for an infinitely small probe at height, h, and located at a distance s from the edge. A more accurate representation of the current distribution near the edges would be a rising exponential, the maximum value of which would again be twice the measured value if there were no contribution from the adjoining side and of the same form as the correction factor (4) but weighted by the exponential. Numerical integration of the weighted correction factor shows that the probe reading at the edge is essentially derived (90%) from currents present up to a distance 3 h from the edge. At the edge, however, the probe does in fact measure currents from the adjoining vertical side according to I ly. h2-h1) fy ..J),. dy dy (5)s 2 1 27T(h -) * .~1 J where I(y) is the current distribution on the adjoining side (the lower half of the y-z planes) and (-y) is the distance measured down from the edge as shown in Fig. 5. The sum total of the contributions to the measured readings there- fore is one half of the sampled current given by (3) plus the sampled current on the adjacent side as given by (5). The cumulative effect of these contributions i -such that the measured currents at the edge are not less than 85% of the actual value. This error translates into an error of less than 3% in computing relative ohmic losses as described by (6). 6] III. Experimental Results g Surface currents were measured for three conductor cross sections (1-1/2 in. by 3 in., 3/4 in. by 3 in., and 1-1/2 in. by 1-1/2 in. ), two conductor orientations, and a range of conductor separations f:om 1/2 inch to infinity. The results are presented in Figs. 6-19 The experimental data for the surface current are normalized to thk total I current flowing in a single conductor such that the integral of the surface current over the four sides A, B, C and D is unity for each of the five I cases investigated. No phase shifts in the current distribution were noted. Because there exists a small contribution from the horizontal current-carrying feed wires, the relative position of the oscillator is noted. I". is tý.sy to observe that this contribution adds to the probe reading on the near side and subtracts on the far side. Typically this asymmetry results in a current on side A with an amplitude approximately 10%6 higher than on side C. From the experimental data presented in Figs. 6-19, proximity effects on ohmic losses may be determined by integrating the square of the surface currents and comparing the results to the appropriate isolated conductor. In so doing, a normalized relative ohmic loss, Ln, may be defined for two identical parallel rectangular cross-section conductors carrying equal currents in terms of a conductor geometry (B/A), relative separation distance (d/C), and referred to the case of electrically isolated conductors. L0(B/A, d/C) Ln(B/A, d/C)II L0(B/A, c) (6) -. ...-- - ... -:[ llU' '4 ~ -"'4 ~~"'4' 4' -- '"' -- - w w z c qT OD-J Vc~J '(0 10 _ z<0 ro <L "C LuL a:U o ZLJH 3cn CC, 0 H) 10 o <0 zZ z <N aCA6r zr(1.NI)~~~u IN iclyD)~l ~Z1J 'I) 4--- -. I CI u w z C3 I r- w co < 0 j *0L 0 0 IL ODD F -f 0 c ~~*u CID z, g S w cz cc 040ICltl< ~~-'x U) 2 cc z < CL, w tn z 0 t 0 0 LLI Lo z 2: 0 U- < WLI 0 u~ < ZIU jo LL (1.NI) .LN~uum 30Viuns G3ZIIVV4HONc U, FF uJ 0-a: wot< z 0w C; 0, (LN- INU)n 30~nOM N oz 01 -% - 0 LL M 00 4 0 0 - z- c/n a:IDOW) (,.l)IN8uo ovun 0ZIVZUO -. --~ --'- --- - I3 cwrq 14 <U '4, 0 ~ 00 cc L HU. Son 0 H Cd, LU W LW 0 W z ~z< -T 1U) _> 0 < K0 0.0 ~0. M 0Z z 0U 0 01 0 0 = = 0 zQ -JJ /C/) Z 9' z U) crp C)0- OD u ---. x f---4 g -. ý*1 a:w w CJ2 U,.) ~w ) Z 0 00 LXXXII1 00%j Z. 0 a:U (L .NI) IN~u-ufl 30Vd~fls O3ZflVVYUONIcZCNJ LLI C',L 0j 00 z 0. )( 0 0 v0c 00L z x x C.')) Coz (L.Nf) 1.N3Hunl ~30VAins G1ZIIVIAJUON w (x: w V o wwf zo Z < ccU< 0-) LU o cr L N3Iuflm 30VJufls G3ZI1V'V4JONU r z IJ V)U -J1z C/) K0 L0') < uUVz U cc -LCo~ cv 3:1: PCt 0 6- 0 0 IL 0 c)x n x) 21 ~ ~ .Nt ±N~Ufl~ 3~d~fS O~flVIN~O IU LL u cc u ~~cr- z z zw c'jZ w -i z<0 Cd) w ,0. 319 z I r- I- I C~)z 0Z LLLD Ž0 .-' CY)0 0 yNI)iN3Ufl3 0 U-rC, z~ w z 0 Z = in- C:) -C 0 <~ a~ a.(I.Nl)INuz)i Z :vdufls 03ZI-IVVINON - _ ~ u0z z z M 0-J4T 0 3: ODO C/,) LU 6 --6x LU, "Ci lot '4U. 0Z 0 1 a a 14 o-- -0 o z~ --Jc << Wo z V (r ZX 0 u 0WDC. (L.N) Ii'~rn~ ~V~fS crZI~v~JuN U U, c I.0 F- 0 -~zZ wi V P q* (1 8 cnw-J 0< zcr-- 0 <~ LL)x< C-) L) ::!z o 0z -J oiL) zz zI 0-: 0 my 6) FI ~ ~ ~ ~ ~ ~ ~ ~ L (LN)i3IfZZXfSOZlvrO u W z i -26- where C, the circumference of one conductor, is held constant. Composite results for L normalized for constarit circumferencen conductors are presented in Fig. 20 where th- dashed lines represent a first-order theory developed in the following section which extends the experimental results to very small separation distances, i. e., d/C < 0. 05. IV. First-Order Theory for Determining Relative Ohmic Losses at the Limits of Separation Distance In analyzing proximity effects for conductors of rectangular cross- section, experimental results show that as the separation distance de- creases the current on the inside face (side D) gradually decreases and is distributed over the remaining sides. At d/C of 0.05 (that is, for example, a 1/2 inch separation of 1-i/2 in. by 3 in. conductors) approximately 95% of the current on the inside face has been redistributed to the other faces (see Figs. 9-11 or Figs. 14-16). Furthermore, for decreasing separation distances the current distributions on sides A and C tend towards the form of the current distribution on the left half of a conductor of dimensions 2A by B (see Figs. 9 or 11, for example). Although it is clear that such a redistribution of current is accompanied by changes in skin depth and varying influences of the corner, it is none- theless useful to construct a simplified model of the process with an assumption that for changes of B/A by a factor of two the relative errors in determining proximity effects which arise from variations in skin depth are minimal and may be neglected. ! i CU~ l- <J, cr 5 r u (n Z U < w j L Il UU < A~~ 1,7 1 1I I I 0 ,to ~Z <iiCI IIi _j 11 m, I *i in C4 0(co LL) 1 0 21 -28- In general when an imperfect conductor is excited by a unit alternating current, it exhibits internal power losses commonly referred to as ohmic losses. Ohmic losses generally are a function of the operating frequency, the geometry of the conductor, its macroscopic properties (which also may be frequency-dependent) as well as proximity effects. In the elementary case of an isolated conductor of unit length, area (h), and resistivity (p) carrying a unit direct current, the ohmic loss (Ld) is given by Ld=A (7) The AC case becomes somewhat more involved because the current is no longer uniformly distributed throughout the conductor but decreases from a maximum at the conductor surface to 1/e of that value at the depth 6, the so-called skin depth. For an isolated imperfect conductor with a constant smooth cross-section (i. e. , no corners or radii of curvature of the same magnitude as the skin depth) and at least several 6 thick, the ohmic loss (Lac) per unit length for a uniformly distributed current with a unit alternating current drive is given approximately by Lac (8) where C is the conductor circumference. Although clearly only an approximation to the actual case, it is greatly simplifying and quite accurate to approximate Lac by (8) when comparing the relative ohmic losses between rectangular conductors of cross-section A by B and ZA by B. This situation is equivalent to a comparison of proximity effects between two rectangular conductors of S-29- 1 dimensions A by B with zero separation distance. In terms of (6) and the above approximation, Ln may be computed at the touching point for two conductors as follows S0 L0(B/A, 0) L ac(B/2A)Ln(B/A, 0) ( 9(B)A, 2 L) (9) L0(BA ac t~~ In terms of (8), L at the touching point may be rewritten asSn 2ZA + B)O- L n(B/A, 0) 2 (10) and reduced to 1+B/IA Ln(B/A,0) -- I+B/A (11) For the values of B/A used, the dashed lines of Fig. 20 represent a first-order theory extending the relative separation distance (d/C) to the limit of 0. For B/A---, it is assumed that B ft C/2 and A $ 28; similarly for B/A = 0, A  C/2 and B f 20. Fig. 21 presents over the range of interest a comparison of the first-order theory with the Cockcroft theory in determining relative ohmic power loss per anit length for parallel rectangular conductors of constant circumference at zero separation distance. Acknowledgements The author is most appreciative to Professor R. W. P. King for sugge;jting the problem and to Dr. D. H. Preis and G. Smith for Z - c0 0 0-j< 0 <c w 0T CN0 < (nw CZU crLL IUJ WUJ cr, 3:z cc 000 02 0 0*~ 0 (0~ V/8 0 -31- providing helpful theoretical and experimental insight. All are members of the Division of Engineering and Applied Physics, Gordon McKay Laboratory, Harvard University. This research was also sponsored in part by the F. J. Seller Research Laboratory, Air Force Systems Command. i I i"I I i -32- I.A References 1. G. Smith, "The Radiation Efficiency of Electrically Small Multi- turn Loop Antennas", Technical Report No. 612, Division of Engineering and Applied Physics, Harvard University, June 1970. 2. J. D. Cockcroft, "Skin Effect in Rectangular Conductors at High Frequencies", Proc. of Royal Society, Vol. 122, No. A790, pp.533-542, Feb. 4, 1929. 3. M. J. 0. Strutt, ''Eddy Currents in an Elliptical Cylinder", Ann. d. Physik, Vol. 84, pp. 485-506, Nov. 2, 1927. 4. W. T. Kelvin, Math. and Phys. Papers, Cambridge University Press, Cambridge, England, 1911, Vol. 5, p. 491. 5. R. W. P. King, "Current Distribution and Impedance per Unit Length of a Thin Strip", IEEE Transactions on Antennas and Propagation, AP-14,No. 2, p. 252, March 1966. 6. For a comprehensive discussion of probe systems see R. W. P. King, it. B. Mack, and S. S. Sandler, Arrays of Cylindrical Dipoles, Cambridge University Press, Cambridge, England, 1968, Chapter 8. -h