Correction to "The Electrically Small Multi-Turn Loop Antenna with a Spheroidal Core"

Ted Simpson' and Yanjie ZhU2

'Uiest of South Carolina, Electrical Engineering DepartmentColumbia, SC 29208, USA

E-mail: ted.simpson@ieee.org

2lIntel Corporation2800 Center Dr., Dupont, WA 98327, USA

E-mail: yanjie.zhu@intel.com

Keywords: Antenna theory; equivalent circuits; electrostatic analysis; ferrite devices; loaded antennas; loop antennas;magnetic cores; magnetostatics; receiving antennas; transmitting antennas

In their recent paper on ferrite-loaded loops, Simpson and Zhu[1] used an approximation for the core loss resistance that was

later found to be invalid for the magnetic-field distributioninvolved. After a brief outline of the problem, the correct lossresistance is derived and its impact discussed.

In [1], the magnetostatic potentials describing the fieldsaround electrically small prolate spheroidal loops were expressedin terms of Legendre functions (altered here for clarity) as

vo 0 AQ, ()P 1 (q), for 4ý4ý( a)

(lb)

The coefficients A and B may be expressed as

A=(NI/2)F,

and

(a, /,U2)

where

F (p] /,UO)

H~i =-Blc. (4)

Using Poynting's theorem [2], it can be shown that if the conduc-tivity and the dielectric losses in the core are neglected, then thereal time-averaged power dissipated in the core may be expressedas a volume integral:

Pi =1 Re2 ffIj.o)re 'Hý" v

(5a)

where ul =,uo (,u,' -j,) is the complex permeability of the core.

Alternatively, the core dissipation may be expressed in terms of anequivalent loss resistance as

P.i, !R,ý1112.dis 2

(2a)

(2b)

(5b)

Since the integrand in the integral in Equation (5a) is constant, theintegral is simply the product of the integrand and the core volumeV. Therefore, from Equations (5a) and (5b), it follows that

Rcoe '/O/4H 1 /112 vH(6)

w u, (2g~r,~N '1 2oi (4No) 2.

P r 3)K2a ýoQI(4O)-piQ(4 o)1

However, in [I] the value of the core loss resistance was given as

and ~oP=(PO1/p')oL,(7)

q (u) =dQi (u)/du.

Since z = c~q, the potential inside the core may be written as

V1= Bz /c. (3)

It follows that the H field inside the core has only a uniform zcomponent, given by

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where p'/,u' is the loss tangent in the ferrite, and L is the induc-tance. However, only under the assumption that all of the magneticflux linking the inductor was in the ferrite medium would Equa-tion (7) be correct. Here, the flux is partly in the core, partly in freespace outside the core, so Equation (7) is clearly not valid.

The efficiency is the only parameter greatly changed by theuse of the correct core loss resistance in Equation (6) rather than

IEEE Antennas and Propagation Magazine, Vol. 49, No. 2, April 2007

O.1

0.01

-I3 E0.1 1 10

alb, length to diameter ratio

Figure 7. The efficiency as a function of alb

cores with a volume of 0.0628 in3.

at 2 NMz for

0T

a ___

1C1r10

a/,lnthtDimee2aiFiur 7a. -The Effcec1 eie sn tecr osrssac

given ~lb byEuain e6nstea tof Equamtion (7).o

Equation (7). This is readily apparent when the earlier version,Figure 7, is viewed beside the revised version, Figure 7a. Only the

efficiency at the highest values of a/b were unaffected by this cor-

rection. It is significant to note that the high-permeability materialsthat also exhibit high loss show the most improvement in effi-

ciency with the indicated alteration in core loss. Also, we note thatconductor (copper) losses dominate at low a/b ratios where, due

to strong demagnetization, the loop behaves as if the ferrite corewere not present. In theory, the current and voltage required toradiate a fixed power that were shown in Figure 11 of [ 1] dependon the core loss as well, and should be affected by the correctiondiscussed. However, this was not the case, since the terminal cur-

rent and voltage depend so strongly upon the inductance of theloop, rather than the loss resistance.

References

1. Ted Simpson and Yanjie Zhu, "The Electrically Small Multi-Turn Loop Antenna with a Spheroidal Core," IEEE Antennas andPropagation Magazine, 48, 5, October 2006, pp. 54-66.

2. Robert Plonsey and Robert E. Collin, Principles and Applica-tions of Electromagnetic Fields, New York, McGraw-Hill BookCompany, 1961.

CorrectionIn the contribution to the AMTA Corner, "Theory and Meas-

urement of Backscattering from RFID Tags," by P. V. Nikitin andK. V. S. Rao [I], Figure 11 was printed incorrectly. The correctfigure, which illustrates the differences between tag antennas withvarious impedance loads, is printed below. For more informationon differential tag radar cross sections, see [2]. The Magazineregrets the error.

References

1. P. V. Nikitin and K. V. S. Rao, "Theory and Measurement ofBackscattering from RFID Tags," IEEE Antennas and PropagationMagazine, 48, 6, December 2006, pp. 2 12-218.

2. P. V. Nikitin, K. V. S. Rao, and R. D. Martinez, "DifferentialRCS of REID Tag," Electronics Letters, 43, 8, April 12, 2007, pp.

IEEE Antennas and Propagation Magazine, Vol. 49, No. 2, April 2007

~0Co.(0a

E0)

Figure 11. The antenna return loss, measured in an anechoicchamber with chip-loaded, short-circuited, and open-circuitedREID tag antennas.

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