Beom-Taek Lee, Emre Tuncer, and Dean Neikirk
The impact of finite interconnect conductivity remains one of the more difficult problems in transmission line analysis. Traditional approaches include the incremental inductance rule and integration over surface current distribution. One difficulty with these techniques is their failure to adequately model the transition from low frequency to high frequency behavior. When calculating broad bandwidth time-domain pulse propagation, for instance, the dispersion induced by this transition can be significant. For digital IC interconnect modeling, several other techniques for predicting the frequency dependence of the series impedance per unit length have also been developed [3-5]. There is a continuing need for models which accurately predict the frequency dependent attenuation and phase constants from dc to high frequency for transmission lines using conductors with finite resistance.
Various methods have been used to calculate the total series impedance of a transmission line structures, such as Weeks' filament method, conformal mapping, and the ribbon technique. Efficient application of such "external" field solvers depends fundamentally upon representing the "internal" behavior of the conductors at the periphery. Arduous calculations can be eliminated by dividing the problem into two steps: first, calculate a quantity which will represent the internal behavior at the surface of the conductors; and second, solve the external problem using this quantity. Similar approaches have been used at high frequencies by applying the surface impedance concept. Although past approaches give good results at frequencies where the skin-depth is much smaller than the cross sectional dimensions of conductors, at and below the frequency range where the skin-depth is comparable to the cross sectional dimensions the conventional surface impedance concept is not applicable. Even at high frequencies, the surface impedance concept is erroneous due to the fact that it does not account for sharp edges. We have developed a new modified internal impedance concept, the Effective Internal Impedance (EII), that is more accurate near sharp edges and covers the transition from high frequency to dc behavior.
EII can be viewed as a quantity which represents the current distribution inside the conductor. At high frequencies, the EII is the same as the surface impedance. The surface impedance can be defined as the ratio of total tangential electric field to total tangential magnetic field at the conductor surface at a given frequency. The surface impedance of an semi-infinite conductor slab is , where u and [[sigma]] are permeability and conductivity of the conductor, respectively. When the thickness of the slab is finite, the surface impedance can be calculated using transmission line equations,
where [[gamma]] is the complex propagation constant in the conductor and d is the thickness. For more useful conductor geometries calculation becomes much more complex. The surface impedance can also be calculated from a conventional full electromagnetic solution, such as one based on the filament method . However, calculating actual surface impedance using such an approach leads to no improvement in numerical efficiency, since the full problem must be solved.
By approximating the surface impedance of a thick rectangular bar with an EII significant speed-up can be obtained. Jingguo et al. calculates surface impedance assuming four plane waves incident normal to the surfaces of the conductor . We have recently developed two new methods that use geometrical segmentation to mimic electromagnetic wave distribution inside the conductor with non-uniform transmission lines . In Figure 1, the EII for the middle sections are calculated using eq. (1) with d=t/2. For the corner sections, each square is divided into two triangles and each triangle further divided into smaller slanted triangles as in Figure 2. For each slanted triangle the input impedance at the base is calculated by
where J0 and J1 are Bessel functions and hn and wn are given as follows,
where N is the total number of slanted triangles and n=0,1,2,...,N-1. For N=4 this model predicts the series resistance of a square bar with in 1% from dc to intermediate frequencies.
A modified version of this method is obtained by making the conductor a hollow tube, with wall thickness of 3d (where d is the skin-depth) as the frequency increases. For the corner regions the same technique is used as that discussed above, while for the rectangular regions the simple hyperbolic tangent expression is used. However, since the thickness of the wall is always 3d, the argument to the special functions (hyperbolic tangents and the Bessel functions in eqs. 1 & 2) is always the same at each frequency once the frequency is high enough such that 3d is smaller than thickness of the conductor. For lower frequencies the dc resistance is used for that section. The impact on numerical efficiency of the EII approach will be illustrated in the next two sections.
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Figure 1: Regions to calculate Effective Internal Impedance; region A uses simple hyperbolic tangent equation (eqn. (1)), and region C' uses eqn. (2).
Figure 2: Each half corner (region C' in fig. (1)) is subdivided into triangles. The equivalent impedance seen at the bottom surface of triangles is calculated by eq. (2).
Figure 3: Conformal mapping process which unfolds conductors in real space (the z-plane), scaling conductor conductivity and surface impedance ZS into the mapped plane (the w-plane).
Figure 4: Comparison between measured (solid lines) and simulated series impedance per unit length (real and imaginary parts) for thin coplanar strips. The strips were supported by a 0.08 cm thick [[epsilon]]r = 2.55 substrate, and the strip dimensions were: d = 0.05 cm, w = 1.3 cm, t = 17 um, and [[sigma]] = 5.8 x 105 S.cm-1 (copper). : conformal mapping results from eq. 10; Z : imaginary part calculated using Maxwell reg.; 8 : real part calculated using Maxwell reg.; : skin effect resistance only (no current crowding).
Figure 5: Series impedance per unit length for copper, coplanar bars, calculated from the volume filament and surface ribbon methods using various surface impedance models. Solid line: volume filament method (from Weeks, );
surface ribbon method, EII from Jingguo et al. ;
: EII from ; ;
EII from skin depth limited simplification of [7, 8] ;
Figure 6: Comparison of volume and surface ribbon calculations for thick, coplanar strips.
Table 1: Comparison of run-times for volume and ribbon calculations of the self-inductance of a rectangular metal bar.