1660 zyxwvutsrqponm IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993 Full-Wave Iterative Variational Formulation for Multiple Coupled Microstrip Lines Ruey-Beei Wu and Zhir-Tsong Hsu Department of Electrical Engincering, National Taiwan University Taipei, Taiwan, Republic of China zyxwvut Abstract zyxwvutsrqpon - zyxw A new variational formulation in terms of two formally decouplctl potentials is derived for the full wave propagation constants of multiple coupled microstrip lines. At very low frequency limits, the present formulation is reduced directly to the conventional quasistatic formulation. The quasistatic solutions are then employed zyxwvutsr as an initial guess to solve the high frequency solution for each coupled mode by a generalized Newton Raphson iterative scheme. Numerical results are presented for two and three coupled microstrip lines of identical or non-identical widths with various spacings and frequencies. I. INTRODUCTION As clock rates increase and inter-line spacings decrease, considerable attention has been paid to the dispersive coupling characteristics of multiple coupled microstrip lines. Most of the analyses are based on the full wave spectral domain approach (SDA), which has been well known for its superiority in dealing with planar transmission lines, especially microstrip lines zyxwvutsrqp [I]. In SDA, Galerkin's method is used to yield a non- linear characteristic equation for the current on the microstrip. The propagation constant can then he found by searching for the root of the characteristic equation. In a system of N coupled microstrip lines, there are in general zyxwvutsrqponm N coupled modes. In case of two microstrip lines with identical width, the two coupled modes can be characterized as even and odd modes, for each of which the total current on each strip can be easily assumed. Then, the SDA can be applied without much difficulty [2,3]. When the coupled microstrip lines have non-identical widths or have more than two lines, the amplitude of the current on each strip is unknown. We need to locate the roots of a determimnt to find the N coupled modes [4-6]. Sometimes, it becomes difficult to search for all these roots satisfactorily, especially vhen the coupling is ncrcrly degenerate. Some approaches have been developed to solve the propa- gation constants withoiit relying on the det,erminaiit sea,rch. Yuan a,nd Nyquist proposed a perturbation approach based on electric field integral equations to deal with two non- identically coupled microstrip lines zyxwvutsr [i']. Thc approach ends up with an N by N eigen-matrix problem, for which an analytic solution is available only when N = 2. Mehalic a,nd Mittra generalized an iteration-perturbation approach in the spatial domain to handle three identically coupled microstrip lines [SI. To start the iterations, they require a given voltage distribution on each strip for the even and odd modes which may not be available for more general cases. Hence, we propose in this paper a new variational formulation a,nd ba,sed on it we give a novel iterative scheme to deal with multiple coupled microstrip lines which may be of different, strip widths. 11. DUAL-POTENTIAL FORMULATION Based on the conventional SDA, it is well known tha,t, the tra.nsformed electric fields and currents on the plane of microstrip lines satisfy a simple algebraic equation [I, (21) on p.3401. Let &(a) = Y[Jx(x)] and jz(a) = Y[Jz(x)] denote the Fourier transform of the transverse and longitudinal currents on the strips, respectively. By the following change of variables Manuscript received June 1, 1992. This work was supported by the National Science Council, R.O.C., under Grant NSC 81-0404-E002-08, it is not difficult to verify that the propagation const.ants satisfy the followiiig variational formula (4) where 9 e and Z e and Z h in SDA by 1 , are related to the TM and TE immittances For example, in the single layer microstrip, where tr and h are t.he relative permittivity and height of the substrate and (9) Note that the dual sources q(x) and zyxw p(x) correspond to electric charge and magnetic dipole, respectively, since Now, the varia.tion formula (3) is formally decoupled for the TM and TE components such that it, becomes possible to solve the dual sources separately. Of course, there a.re some boundary conditions for the dual sources since t,he currents Js(.x) = Jz(x) = 0 on the slot re ions other than strips. It can be readily found from (I) and (27 that on each slot region. dual sources should satisfy the constraints q(x) = 0, p(x) = some comtant (11) In addition, the difference between t,lie 1: [p(x) - q(x) - p(xt)/ cosh[/3(z~-x)] = 0 (12) P(Xd - Pht) 22 = P. 1 .T1 [p (T) - q(x) - p (~111 sin~/i~j~x~ -.[)I d.i- ( 13) where md xz are two arbitrary point: on adjacent slot. regions. 111. QUASI-STATIC LIMIT At bhe quasi-static limit. It is not difficult to verify that 9 zyx p=O) is just the Fourier trmsform of the elect,ric potential produced by a line, charge. Since /3 /. is the charge on the strips as explained in (lo), We in zyxwvut (4 can be physically interpreted as the electric energy stored in the microstrips per unit length. In other wolds, if Q,, (1 5 n 5 N) is the total static charge on the n th strip. I[('? 1;- 0, p=X:.Jc,ti+ 0, lil+ /cv/, K ~ + /e/. (14j 0018-9464/93$03.00 0 1993 IEEE