square conductor fields inside

Progress In Electromagnetics Research M, Vol. 23, 139{152, 2012 IMPROVED NUMERICAL METHOD FOR COMPUT- ING INTERNAL IMPEDANCE OF A RECTANGULAR CONDUCTOR AND DISCUSSIONS OF ITS HIGH FRE- QUENCY BEHAVIOR M. Matsuki and A. Matsushima* Graduate School of Science and Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan Abstract |An e±cient numerical solution is been developed to compute the impedances of rectangular transmission lines. Method of moments is applied to integral equations for the current density, where the cross section is discretized, to improve the convergence, by a nonuniform grid that obeys the skin e®ect. Powerfulness of this approach up to rather high frequencies is veri¯ed by comparing with asymptotic formulas and other references. Detailed discussion is given for the current density distribution and its e®ect to the impedance, especially for a high frequency range. 1. INTRODUCTION Transmission lines having conductors of rectangular cross section are typically found in MMIC [1, 2], and lots of formulas for the characteristic impedance and the attenuation coe±cients have been proposed and collected [3, 4]. In the recent trend of high speed digital signal processing, the increase in the per-unit-length resistance and inductance due to the skin e®ect has become an issue of great interest. This problem, especially for a free-standing rectangular conductor as a basic structure, has been discussed from the viewpoints of analytical treatment [5{9], experiment [10], and numerical computation [11{17]. The classical approach based on the conformal mapping and perturbation [5{7] predicts the high frequency asymptotic behavior of the per-unit-length resistance in terms of the surface resistance and the complete elliptic integrals in the exact manner. Full-range Received 21 December 2011, Accepted 23 January 2012, Scheduled 1 February 2012 * Corresponding author: Akira Matsushima ([email protected]). 140 Matsuki and Matsushima solution is attainable only by proper numerical techniques, and some of them [2, 12, 15, 16] discussed the accuracy of the obtained impedance values when the skin depth is much smaller than the line thickness. Nevertheless, it would seem that no attempt has been made to bridge the above mentioned analytical and numerical results. Besides, the series solution in [9] based on the Dirichlet boundary condition cannot incorporate the singular behavior of the current density at four corners. Taking account of the above aspects, we ¯rst improve the computational method for the internal impedance of a rectangular conductor, and next give a circumstantial discussion about its high frequency behavior by comparing with analytical solutions. The integral equation for the current density in the cross section is solved by the method of moments [18]. Unlike the present author's approach [19], where the line is replaced by a set of round wires and the current is expanded by cylindrical functions, we here utilize simple rectangular segments and a constant basis. This process is, in principle, similar to that employed in [2, 13{15, 17], and among them, [15] proposed the use of a nonuniform grid without detailed explanation. The present paper, however, describes how to construct such grids systematically in accordance with the skin e®ect. The e®ectiveness of this treatment is demonstrated by numerical experiments. Referring to the perturbation solution [5{7] and the ¯tted formula [8], we gain new insights into the high frequency properties of the impedance and current density distribution. 2. NUMERICAL METHOD 2.1. Integral Equations As shown in Figure 1, a conductor transmission line with rectangular cross section is free-standing in the vacuum. The cross section of the line is expressed by S=fr(=ixx+iyy)j0< x < w; 0< y < t g, where xy 0t( , ) 0 wSε µ 0 ( , )σ µ 0 Figure 1. Rectangular conductor line in the vacuum. Progress In Electromagnetics Research M, Vol. 23, 2012 141 ixandiyare the unit vectors in each direction. The conductivity of the line is ¾, and the permittivity of the vacuum is "0. The line is assumed to be nonmagnetic, and so the permeability is ¹0everywhere. The permittivity of the line is not taken into account, because we deal with the metal line where the displacement current is neglected. Suppose that the current density J(r) in the line has only the axial component Jz. Then the vector potential A(r) has the same component which satis¯es the Helmholtz equation in the vacuumµ@2 @x2+@2 @y2+¯2 0¶ Az(r) = 0 ; ¯2 0=!2"0¹0 (1) Under the time factor ej!t, the solution of (1) is written in terms of the Hankel function of the second kind as Az(r) =¹0 4jZ S[SrJz(r0)H(2) 0(¯0jr¡r0j)dS0(2) where r0=ixx0+iyy0,dS0=dx0dy0, and Sris the cross section in which the return current °ows. Let us ¯x Srat such a far position that the signal current is not a®ected by it. We also assume that jrj is much smaller than the vacuum wavelength 2 ¼=¯ 0. This situation allows us to replace H(2) 0(¯0jr¡r0j) with ( ¡2j=¼)(logjr¡r0j+C0) (C0: constant) and H(2) 0(¯0jr0j) for SandSr, respectively. Then we have Az(r)¼ ¡¹0 2¼µZ SJz(r0) logjr¡r0jdS0+IC0+Cr¶ (3) where the total current Iand the constant Crare written as I=Z SJz(r0)dS0; C r=j¼ 2Z SrJz(r0)H(2) 0(¯0jr0j)dS0(4) Employing the relations concerning the electric ¯eld that E= ¡j!A¡ rÁ=J=¾, with Ábeing a scalar potential, we are led to the set of integral equations for the current density as Jz(r)¡j ¼±2Z SJz(r0) logjr¡r0jdS0=C(r2S) (5) where ±=p 2=(!¹0¾) is the skin depth of the metal, and C= ¡¾(@Á=@z )S+j(IC0+Cr)=(¼±2) is unknown constant. 2.2. Method of Moments Let us discretize (5) by the method of moments [19], where two dimensional rectangular pulse functions are used for both basis and weighting functions [15, 17]. 142 Matsuki and Matsushima First, we slice the cross section SbyM+ 1 horizontal lines and N+ 1 vertical lines½ x=x0(= 0) ; x 1; : : : ; x m; : : : ; x M¡1; x M(=w) y=y0(= 0) ; y 1; : : : ; y n; : : : ; y N¡1; y N(=t)(6) such that S=SM m=1SN n=1Smnwith Smn=frjxm¡1< x < xm; yn¡1< y < y ng. The subsection Smnhas the area ¢ Smn= ¢wm¢tn, where ¢ wm=xm¡xm¡1and ¢ tn=yn¡yn¡1. Next, the current density Jzis approximated by a constant Jmn inSmn. This procedure leads us to the set of linear relations Jmn+MX m0=1NX n0=1Gmn;m0n0Jm0n0=Cµ m= 1;2; : : : ; M ; n= 1;2; : : : ; N¶ (7) where Gmn;m0n0=¢Sm0n0 j¼±2logRmn;m0n0 (8) and the symbol Rmn;m0n0is a geometrical mean distance (GMD) between SmnandSm0n0de¯ned by logRmn;m0n0=1 ¢Smn¢Sm0n0Z SmnZ Sm0n0logjr¡r0jdS0dS (9) The analytical expression of GMD is given in [20, 15]. Since the constant Cin (7) is unknown, one condition is lacking. This is supplemented by discretizing the former expression in (4) as MX m=1NX n=1Jmn¢Smn=I (10) where Iis a preset constant. Equations (7) and (10) are altogether solved numerically for MN + 1 unknowns JmnandC. 2.3. Impedance Calculation The per-unit-length resistance and internal inductance of the line are obtained in terms of Joule heat and stored magnetic energy, respectively. They are written as R=1 ¾jIj2Z SjJ(r)j2dS¼1 ¾jIj2MX m=1NX n=1jJmnj2¢Smn (11) Lin=¹0 jIj2Z SjH(r)j2dS¼¹0 jIj2MX m=1NX n=1jH(¹rmn)j2¢Smn (12) Progress In Electromagnetics Research M, Vol. 23, 2012 143 where ¹rmn(=ix¹xm+iy¹yn) is the midpoint of Smnsuch that ¹ xm= (xm+xm¡1)=2 and ¹ yn= (yn+yn¡1)=2. The magnetic ¯eld in (12) can be computed by applying (3) into H= (1=¹0)r £Aas H(r) =1 2¼MX m=1NX n=1Jmn[¡ixK(¢tn;¢wm; y¡yn; x¡xm) +iyK(¢wm;¢tn; x¡xm; y¡yn)] (13) where the double integral K(a; b; x; y ) =Zb=2 ¡b=2Za=2 ¡a=2x¡x0 (x¡x0)2+ (y¡y0)2dx0dy0(14) is evaluated analytically as described in Appendix A. 3. NUMERICAL RESULTS 3.1. Convergence Let us develop an e®ective solution to the integral equation by contriving computational grids as follows. Uniform grid The simplest way is to ¯x as ¢ wm= ¢w=w=M and ¢ tn= ¢t=t=N with keeping M=N ¼w=t, which gives xm=m¢wandyn=n¢t. But this is ine®ective at high frequencies where the skin e®ect becomes prominent. Nonuniform grid Better convergence is expected if we let the segments smaller near the conductor surface than at the center. Accordingly, let us set the position of the nodes xmby the rule m M=Zxm 0jgw(x)jdxÁZw 0jgw(x)jdx(m= 0;1; : : : ; M ) (15) where the function simulating the current behavior is gw(x) = cosh(1 +j)(x¡w=2) ±Á cosh(1 +j)w 2±(16) so that gw(0) = gw(w) = 1. Its amplitude in the high frequency region is approximated as jgw(x)j ¼coshx¡w=2 ±Á coshw 2±(±¿w) (17) Solving (15) and (17) leads us to xm=w 2+±arcsinh·2m¡M Msinhw 2±¸ (18) 144 Matsuki and Matsushima Similar procedure gives us the node ynin the form analogous to (18), with w,m, and Mreplaced by t,n, and N, respectively. The balance of the truncation numbers is kept by M N¼Zw 0jgw(x)jdx Zt 0jgt(y)jdy=tanhw 2± tanht 2±(19) Figure 2 shows the examples of grid at middle and high frequencies f(=!=(2¼)). Note that the skin depth ±is 20.9 ¹m and 2.1 ¹m at 10 MHz and 1000 MHz, respectively. In Figure 2(a), ±andtare comparable so that M > N . However, in Figure 2(b), the ratio ±=t= 0:09 is so small that the criterion (19) predicts M=Neven for the rectangular cross section. Figure 3 shows the real and imaginary parts of the line impedance computed from (7) and (10) as a function of the truncation number (a) (b)f = 10 MHz, M = 7, N = 3 f = 1000 MHz, M = 7, N = 7 Figure 2. Grid for w= 5t= 111 :8¹m and ¾= 58 MS/m. (a) (b) Impedance [ Ω/m] Nonuniform gridM 1 M__50 40 20 10 5 R Linω f = 1000 MHzw = 5t = 111.8 m µNonuniform grid Uniform gridImpedance [ Ω/m]M 1 M__50 40 20 10 5 R Linω f = 1000 MHzw = t = 50 m µ 0 0.05 0.1 0.15 0.2010203040506070 0 0.05 0.1 0.15 0.22030405060 Figure 3. Convergence of per-unit-length impedance for copper lines having common parameters wt= 2500 ¹m2and¾= 58 MS/m. (a) Square line. (b) Rectangular line. Progress In Electromagnetics Research M, Vol. 23, 2012 145 M. The converged values of Rand!Linare found by extrapolating the curves up to the vertical axis where M! 1 . Though the sectional area wtis common, the resistance Rconverges to di®erent values, 52 ­/m and 43 ­/m, for Figures 3(a) and (b), respectively. This di®erence stems from the fact that, at high frequencies, Ris inversely proportional to the perimeter 2( w+t) at a rough guess. Comparison of the solid and broken lines in Figure 3(a) reveals the e®ectiveness of using the nonuniform grid shown in Figure 2. There are two independent series for odd Mand even M, and the former exhibits better convergence. 3.2. Frequency Characteristics Figure 4 shows the frequency dependence of the impedance compared with other solutions. Under the ¯xed cross section wt, the ratio w=t is chosen as 1 (square), 5, and 10 for Figures 4(a), (b), and (c), respectively. The solid curves are the numerical solution to the linear Equations (7) and (10). The resistance Rand reactance !Linbehave asO(f0) and O(f1), respectively, at low frequencies, whereas, they shift to the same order O(f1=2) as frequency increases. The curves for Ragree well with the open circles based on the previous reports [15, 8]. In [15] the method of moments similar to ours is used, and in [8] the expression R Rdc=8 >>< >>:0:43093 xw 1 + 0 :041(w=t)1:19+1:1147 + 1 :2868xw 1:2296 + 1 :287x3w +0:0035( w=t¡1)1:8(xw¸xw0) 1 + 0 :0122x3+0:01x2 ww (xw< xw0)(20) is proposed by ¯tting the measured values in [10]. Here, the dc resistance, the parameter xw, and its boundary are Rdc=1 ¾wt; x w=r 2 ¼¢p wt ±; x w0= 2:5 (21) The crossover frequency corresponding to xw= 2:5 is 17.1 MHz, beside which the sequence of circles exhibits a slight discontinuity in Figures 4(b) and (c) due to the separate expressions in (20). These could be smoothly bridged by choosing the boundary xw0not as constant 2.5 but as xw0=8 < :4:5¡1:4p w=t (1< w=t < 4) 1:7 (4 < w=t < 9) ¡0:1 + 0 :6p w=t (9< w=t < 16)(22) which varies between 1.7 and 3.1. 146 Matsuki and Matsushima (a) (b) (c) (d)LinωRf [MHz] f [MHz]Impedance [ Ω/m]R Linω f [MHz]Impedance [ Ω/m] LinωRNumerical (7)(10) Asymptotic (23)(28) Reference [15] 10 100 10000.70.80.91 f [GHz]Lin / R ω Asymptotic (23)(28) Reference [16]w = 5t = 111.8 = 58 MS/mσ w = 10t = 158.1 = 58 MS/mσ w = 10t/3 = 20 = 30 MS/mσw = t = 50 = 58 MS/mσ60 1 10 100 1000131030 60 1 10 100 10001310301 10 100 100013103060 Impedance [ Ω/m]Numerical (7)(10) Asymptotic (23)(28) Reference [8] Numerical (7)(10) Asymptotic (23)(28) Reference [8]µm µm µm µm Figure 4. Frequency dependence of the per-unit-length impedance. The parameters wtand¾are common to (a), (b), and (c). (a) w=t= 1. (b)w=t= 5. (c) w=t= 10. (d) Ratio of reactance to resistance for w=t= 10 =3. In [16], the line is put on a dielectric substrate and is surrounded by an enclosure made of electric and magnetic walls. Let us shift our subject to the high frequency behavior in Figure 4. It is well known that the conductor with circular cross section has the property that the ratio !Lin=R!1 asf! 1 . This is not the case, however, for rectangular lines, and discussions about the behavior of !Lin=Rhave ever occurred [9, 15, 16]. It deserves to reconsider this problem, and therefore, we quote the following asymptotic expressions forRand!Lin. Resistance The charge distribution on the conductor surface is Progress In Electromagnetics Research M, Vol. 23, 2012 147 derived by the conformal mapping method for the static case. Introduction of the surface resistance Rs= 1=(¾±) allows us to write the formula as [5{7] R=2Rs ¼2p wtp [E(·)¡~·2K(·)][E(~·)¡·2K(~·)] [K(·)+K(~·)] (23) where K(¢) and E(¢) are the complete elliptic integrals of the ¯rst kind and second kind, respectively, and ~ ·=p 1¡·2. The value of·is a solution of t=w=£ E(·)¡~·2K(·)¤±£ E(~·)¡·2K(~·)¤ (24) In the special case where w=t, we have ·2= ~·2= 0:5 and R=Rs=(¼w). Internal inductance We intentionally model the magnetic ¯eld as H(r)¼I 2(w+t)[¡ht(y)ix+hw(x)iy] (r2S) (25) circulating inside S, where hw(x) = sinh(1 +j)(x¡w=2) ±Á sinh(1 +j)w 2±(26) so that hw(0) = hw(w) = 1. Using J=r £H, we can con¯rm that (4) is satis¯ed if w; tÀ±. Substituting (25) into (12) and employing the high frequency expression of jhw(x)jsimilar to (17), we are led to the asymptotic formula !Lin¼Rs 2(w+t)=1 2(w+t)r ¼¹0f ¾(27) This gives rather accurate values if w=t, but this is not necessarily the case for rectangular cross sections because of the di®erent ¯eld distribution near the longer and shorter sides. Via numerical experiment, we improve the accuracy by modifying the frequency dependence as !Lin¼1 2(w+t)s ¼¹0f0 ¾µf f0¶(w=t)0:05 (w¸t) (28) with the reference frequency f0= 19=(¼¹0¾w0:5t1:5). In Figures 4(a){(c), we ¯nd that the results by the asymptotic formulas (23) and (28) approach the present numerical solution of (7) and (10) as frequency is increased. Agreement is observed roughly asf > 200 MHz corresponding to ±=p wt < 1=10. Figure 4(d) 148 Matsuki and Matsushima demonstrates the validity of the asymptotic formulas by comparison with the full-wave mode matching approach in [16]. Even at f= 1000 GHz that corresponds to ±=t¼1=100, the ratio !Lin=Ramounts to 0.9 and is far from 1. Note that, though the geometries are not exactly the same, such a comparison still makes sense. This is because the ratio !Lin=Ris considered to be rather insensitive to the width of enclosure (now w+ 2s= 35 ¹m) introduced in [16]. In fact, as shown in Figure 5 of [18], the proximity e®ect due to the electric walls and line images adds to RandLinin similar extent. Besides, the existence of dielectric substrate does not a®ect the pattern of the magnetic ¯eld. 3.3. Current Behavior Figure 5 shows the current density distribution in the square cross section at a medium frequency such that w=t= 5:4±. The density Jz(x; y), normalized by the dc value I=wt, are displayed by contour lines. The skin and edge e®ects are discussed as below. ²Figure 5(a) is based on the numerical solution of (7) and (10). The contours are close to concentric circles and the current concentrate into four corners with the maximum value 3.5. The impedance is 12:7 +j9:5 ­/m. The shapes and arrangement of contour lines resemble those of Figure 3(b) in [12]. ²It is meaningful to mention the analytical solution under the Dirichlet boundary condition [9]. A constant voltage source is impressed over the rectangular surface, by which the per-unit- length impedance and the internal electric ¯elds are obtained as the series expressions (10) and (15) in [9], respectively. As shown in Figure 5(b), this boundary condition gives constant current density 2.1 along the pro¯le. Though this approach works for the circular cross section, it fails to take into account the edge e®ect of rectangular conductors. This is why the impedance, 11:6 +j9:0 ­/m, is a little lower than that of Figure 5(a). ²To give suggestive information, we take up the plausibility of (25) representing the magnetic ¯eld H. Figure 5(c), drawn by J= r£Hcombined with (25), clearly represents the edge e®ect. The de¯nition of resistance (11) leads us to R¼!Lin+2 ¾(w+t)2(29) with !Lingiven in (27). The impedance 12 :6 +j9:3 ­/m is comparable to that of Figure 5(a). Note that this approach gives rather accurate internal inductance as f¸50 MHz, but that the resistance at 1000 MHz is underestimated by 14% compared with Progress In Electromagnetics Research M, Vol. 23, 2012 149 (a) (b)2 2 2 21 1 01020304050 0 10 20 30 40 5022 21 1 01020304050 0 10 20 30 40 50 x [µm]y [ m] 3 33 3 y [ m] 2 2 2 211 01020304050 0 10 20 30 40 50 (c)y [ m] (b) (c)R + j ωL [Ω]in (a) 12.9 + j9.5 11.6 + j9.0 12.6 + j9.3µ µµx [µm] x [µm] Figure 5. Normalized current density jwtJz(x; y)=Ijforw=t= 50¹m,¾= 58 MS/m and f= 50 MHz. (a) Present numerical solution of (7) and (10). (b) Based on the Dirichlet boundary condition [9]. (c) Based on the approximate magnetic ¯eld (25). the numerical result. Needless to say, the above discussion holds only for the square cross section. After the manner of [2] and [12], we evaluate the coe±cient kwhich indicates how the current concentrates into the conductor surface. The results are listed in Table 1 for several values of w=tand 2 ±=t. The value ofkis de¯ned as follows. As in Figure 4.23 in [2] or Figure 10 in [12], we suppose that the current °ows uniformly along the conductor surface 150 Matsuki and Matsushima Table 1. Coe±cient k=Rs=[R(w+t)] indicating the degree of concentration of the surface current. Present solutionPPPPPPPP2±=tw=t1 2 4 8 1 0.96 1.17 1.20 1.17 0.5 1.40 1.43 1.41 1.32 0.25 1.50 1.49 1.43 1.32 +0 1.57 1.55 1.44 1.32 Reference [2] Reference [12] PPPPPPPP2±=tw=t1 10.7 ·0:7 1.3 0.286 1.486 0.264 1.58PPPPPPPP2±=tw=t·6 small 1.67 with the thickness k±=2. This estimates the approximate resistance as R=1 ¾¢2(w+t)¢(k±=2)=Rs k(w+t)(30) with Rs= 1=(¾±) being the surface resistance. From (30) we have k=Rs=[R(w+t)]. In Table 1, the square case at w=t= 1 shows the largest variation in kaccording to the change of 2 ±=t. This is because the current shifts from °at surfaces to separated four corners, making kunstable. Results in [2] and [12] are also tabulated, where kextends from 1.3 to 1.67 for several parameters ( ·in [12] is about 1.2, which corresponds to 2 =kin the present notation). The maximum values of kcan be predicted by the asymptotic formula (23), and are given in the lowest row for 2 ±=t= +0 in the present solution. If w=t= 1, (23) and (24) give the analytical result that ·2= 0:5,R=Rs=(¼w), and k=¼=2 = 1 :57. The data at 2 ±=t= +0 are ¯tted by the formula k= 1:57¡0:06 [log( w=t)]2(31) the absolute error of which is less than 0.01. Finally, in order to ¯nd out the relation between the independent descriptions in [2] and [8], let us examine the crossover frequency at which the resistance curve shifts from O(f0) toO(f1=2). As done in [2], we set the high frequency resistance Rin (30) equal to the dc value Rdcin (21), leading to wt=k±(w+t). This equation is reduced to w=¼±ifw=tandk=¼=2. Then the parameter xwin (21) becomesp 2¼¼2:5 which agrees with the value proposed in [8]. Progress In Electromagnetics Research M, Vol. 23, 2012 151 4. CONCLUSION The numerical solution has been developed to compute the impedances of rectangular transmission lines. Method of moments was applied to integral equations for the current density, where the cross section is discretized by a nonuniform grid to improve the numerical e±ciency. Comparison with other data reveals the powerfulness of the present approach for wide frequency ranges. Special attention is paid to the current density distribution and its e®ect to the impedance. The present analysis can be readily extended to the cases where the number of conductor is increased or the conductors are attached on a lossy substrate. These problems deserve further treatment. 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