IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 4, JULY 1991 3601 Demagnetizing Factors for Cylinders Du-Xing Chen, James A. Brug, Member, IEEE, and Ronald B. Goldfarb, Senior Member, IEEE Abstract-Fluxmetric (ballistic) and magnetometric demag- magnetic poles at each end of a cylinder [ 191. This sim- netizing factors Nf and N, for cylinders as functions of suscep- plistic model could be used only for long uniformly mag- tibility x and the ratio y of length to diameter have been eval- netized cylinders, and the results deviated significantly uated. Using a one-dimensional model when y 2 10, Nf was from experimental data on ferromagnetic samples. During calculated for - 1 5 x < Q) and N,,, was calculated for x + 00. Using a two-dimensional model when 0.01 5 y 5 50, an im- the 1920’s and 1930’s, there were several theoretical pa- portant range for magnetometer measurements, N, and Nf were pers on Nf for material with constant susceptibility x. The calculated for -1 5 x < 00. Demagnetizing factors for x < 0 results were given as functions of x and the length-to- are applicable to superconductors. For x = 0, suitable for diameter ratio y. These used one-dimensional models with weakly magnetic or saturated ferromagnetic materials, Nf and N,,, were computed exactly using inductance formulas. approximations as needed to suit the computational tech- niques of the time. The first systematic theoretical cal- culation of N’ for the high susceptibility case was done by I. INTRODUCTION Wurschmidt [20], [21]. He calculated Nf of cylinders HE study of demagnetizing factors for ellipsoids and using a one-dimensional model in which the cylinder had Ttheir degenerate forms (spheres, infinite plates, infi- side surface poles and point end poles. He used Taylor nite cylinders) dates from the work of Poisson [ 11 and was expansions for the magnetization and the demagnetizing elaborated upon by Thomson [2], Evans and Smith [3], field at the midplane. The calculation was complicated, and Maxwell [4]. Experimental investigations on finite and he completed it only for the case y = 50 and x --+ W. right circular cylinders began in the 1870’s, when ballistic For the y and x dependence of Nf,he gave qualitative galvanometers were first used in magnetic measurements results using the first few terms of the expansion. A sim- of iron [5], [6]. The literature distinguishes between ilar approach with simpler expressions was used by Neu- “magnetometric” and “fluxmetric” (or “ballistic”) de- mann and Warmuth [22], who calculated Nf for x + 00 magnetizing factors N, and Nf [7]. N, refers to an average andy 2 10. of magnetization over the entire specimen and is appro- To obtain the susceptibility dependence of Nf,Stablein priate for magnetometer measurements of small samples. and Schlechtweg [23] used a quadratic approximation and Nf refers to an average of magnetization at the midplane two linear differential equations. The model was im- of the sample and is appropriate for measurements made proved by substituting uniform end-surface poles for point with short search coils. Values of the demagnetizing fac- end poles. Their results included 30 values of Nf for 10 tor were deduced from the shearing of the magnetic hys- I y 5 500 and 12.56 Ix c W.An extension of the y teresis loop [8], a procedure originally developed by Lord region to 0 was achieved by Warmuth [24]-[26], who fit- Rayleigh for ellipsoids [9], or by measuring the magnet- ted existing data and extrapolated graphically using the ization and the field at the cylinder’s side [ 101. Factors demagnetizing factor N of ellipsoids as a reference. The for cylinders of several aspect ratios were published [7], values of Nf calculated from the one-dimensional models [8], [ll], [12]. By the 1900’s, it was apparent that the were consistent with the data of ballistic measurements values of the factors depended on the susceptibility x of on soft magnetic materials. Bozorth and Chapin [27] com- the material [13]-[15]. piled the results, which were later plotted in Bozorth’s Although it has been criticized from a pedagogical point book [28]. of view [16], [17], the use of fictitious magnetic poles to To obtain axial demagnetizing factors more accurately, calculate demagnetizing fields has been universal. The especially for short cylinders, two-dimensional calcula- first theoretical treatment of magnetic pole distributions tions are needed. The simplest case is x = 0, where N, in finite cylinders was by Green [18]. An early model to and Nf as functions of y can be derived analytically. The attempt to explain experimental data considered point approximation x = 0 applies to diamagnets, paramagnets, and saturated ferromagnets. N,,, for 25 values of y from Manuscript received March I I, 199 1, The authors are with the Electromagnetic Technology Division, National 0.2 to 1000 were obtained accurately to four significant Institute of Standards and Technology, Boulder, CO 80303. figures by Brown [29] from a calculation of self-induc- D.-X. Chen is on leave from the Electromagnetism Group, Physics De- tance [30] and listed as a table in Brown’s book [31]. partment, Universitat Autbnoma de Barcelona, 08 I93 Bellaterra, Spain. J. A. Brug is on leave from the Thin Film Department, Hewlett-Packard Crabtree [32] obtained the same values for the average Laboratories, Palo Alto, CA 94303. demagnetizing factor by integration of the local field over IEEE Log Number 9102083. the cylindrical volume. Moskowitz et al. extended U.S. Government work not protected by U.S. Copyright 1.- i-r 3602 IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 4, JULY 1991 Brown’s method to cylinders of polygonal cross section ders. Brug and Wolf [57] calculated the magnetization [33], and KaczCr and Klem extended it to hollow cylin- distribution in disks and obtained the local demagnetizing ders [34]. Nf for x = 0 was calculated exactly by Joseph factor for materials that undergo phase transitions. [35]. Approximate values for N, and Nf for x = 0, ac- In Zijlstra’s book [58], Nf and N, are plotted. These curate for large y, were calculated by Vallabh Sharma types of graphs and tables appear in other books on mag- using uniformly magnetized volume elements [36]. Sato netism and magnetic materials, and they are widely used, and Ishii [37] obtained a simple expression to approxi- sometimes inappropriately, in magnetic measurements of mate N, for x = 0. Chen and Li [38], [39] obtained Nf ferromagnetic, ferrimagnetic, weakly magnetic, and su- for x = 0 using magnetostatic potential calculations. perconducting materials. However, there remain some The susceptibilities x = - 1 and x -+ 00 correspond to problems. For x = 0, the most accurate case, the number perfectly diamagnetic and ideally soft ferromagnetic ma- of y values for N, and Nf is insufficient for accurate in- terials, respectively. N, and Nf for these susceptibilities terpolation. For x # 0, almost all books give results ob- were first treated by Taylor for perfectly conducting cyl- tained before 1950, and there are no data for x < 0. For inders [40], [41]. He developed a method introduced by long cylinders (y > lo), there is a lack of data on the x Smythe that expressed charge densities on the side and dependence of Nf,and there are no data on N,. For short ends in terms of a set of orthogonal polynomials, and ex- cylinders (y < lo), there are even less data, and those panded the electrostatic potential at the cylinder center that exist have large errors because they were obtained by [42]-[44]. Taylor calculated electric and magnetic polar- extrapolation. In summary, there is no complete picture izabilities for conducting cylinders for 0.25 Iy 5 4 in for the y and x dependence of Nf and N,. both the longitudinal and transverse directions. N,(m) can In this paper, we calculate Nf and N, for a complete be deduced from his electric polarizability results because range of y and x. Susceptibility x is traditionally assumed of the analogy between electrostatics and magnetostatics. to be constant in the material and is therefore defined as Because his calculation for magnetic polarizability was M/H, where M is the magnetic moment per unit volume for a uniform quasi-static but nonpenetrating applied field, and H is the internal magnetic field. For the case x = 0, N,( - 1) can also be deduced from his results. According in which the magnetization is uniform, we give 61 exact to Taylor, his convergence error was less than 0.1 % for inductance calculations of N, and Nffor lop5I y 5 lo3. the longitudinal direction. For x # 0, more elaborate methods are used. For y > Using a similar approach with simpler base functions, 10, the variation of magnetization across the radius of the Templeton et al. calculated axial Nf for x -+ 03 for 0.05 cylinder is negligible at the midplane, and we calculate I y I 250 [45], [46]. The fact that the side and end- Nf as a function of y and x (- 1 Ix < 00) based on the pole densities have basically a 6-’13 dependence, where one-dimensional model of Stablein and Schlechtweg [23]. 6 is the distance from the corner, was used to construct Unlike them, we use Taylor expansions for M(z), calcu- the set of polynomials. To estimate their error, Templeton late the demagnetizing field Hd(z) directly at 25 points and Arrott calculated the root-mean-square deviation of along the axis, and obtain more accurate results. The the normalized potential from 0 and found it to be less model is also applicable to N, for x -, 00. For 0.01 Iy than 0.31 % [45].