1 Introduction In modern high-speed printed circuit boards, in order to shorten the interconnection lines between electronic components, the multi-layer packaging structure has been widely used. The interconnection lines used in the multi-layer packaging structure include not only microstrip lines or striplines in the traditional sense at each level, but also vertical transmission lines for connecting signal lines at various levels, so-called "through holes." Via is one of the most commonly used interconnects in printed circuit boards. Because it is a "discontinuous" structure in the entire interconnect, the high-frequency component of the high-speed signal will be attenuated when passing through the via. Reflection causes distortion of the signal, at the same time, coupling and crosstalk will also occur between the through holes, which brings a series of signal integrity problems and affects the transmission of the signal. Therefore, it is particularly important to model its structure and analyze the transmission characteristics.
The main methods for analyzing the through-hole structure are the quasi-static method for low-frequency conditions and some numerical methods for high-frequency conditions, such as the moment method (MoM) received on August 23, 2005, and changed back to Anhui in 2006-02-20 The Provincial Department of Education Key Research Fund (2004kj002zd) funded projects and the finite-difference time-domain method (FDTD). Although the numerical method has high precision and can theoretically simulate the transmission and coupling characteristics of through holes, it consumes a lot of computer memory and calculation time. Therefore, in practical applications, these methods generally can only simulate a part of the entire package interconnect structure . Documents <5, 6> proposed an equivalent network model method (NW). For the two-hole coupling problem, this method assumes that the two vias are completely symmetrical and ignores the effects of other vias around it, so it cannot Really simulate the transmission characteristics of the actual through hole, and this method is only suitable for high frequency below 20GHz.
In this paper, we first model a single-layer via structure and use a semi-analytical full-wave analysis method to solve the Foldy-Lax multipath scattering equation between cylinder vias in a planar waveguide composed of two ideal conductive plates To get the corresponding scattering parameter matrix. Then the single-layer via structure is regarded as a multi-port network. According to the multi-port network cascade theory, the scattering parameter matrix of the multi-layer via is obtained. Finally, the scattering parameters of the four-layer double-hole are calculated, and the calculation results are in good agreement with the calculation results of the literature <6>. This paper also gives the calculation results of the scattering parameters of the excited double-hole in the four-layer multiple cylindrical through holes. It is shown that the scattered vias distributed around have little effect on the transmission parameters of the excited double holes, but will have a greater impact on the coupling coefficient.
2 Modeling of vertical vias 2.1 Through-hole structure Taking a single through-hole as an example, as shown in Figure 1 (a), the entire through-hole structure is composed of 3 parts: the signal lines on the upper and lower conductive plates and the A transmission line or a vertical through hole connecting these two signal lines and passing through holes in two ideal conductive plates. Between the conductive plates and the surface of the two conductive plates is a medium with a relative dielectric constant of rε. According to the principle of equivalence, the holes on the ideal conductive plate can be filled with ideal conductors, and an equivalent magnetic current of equal size and opposite direction is added to both sides of the filled ideal conductor. It can be decomposed into internal problems and external problems to deal with separately. Finally, the two parts are connected according to the connection condition that the equivalent magnetic currents are equal in size and opposite in direction, and the entire through-hole problem can be solved.
To deal with the method introduced in this article, this article mainly discusses the internal problems of the through hole, that is, the modeling and analysis of the vertical through hole part.
2.2 Modeling of single-layer vertical vias Single-layer vertical via structures. Assuming that two ideal conductor plates are located at / 2 zd and / 2 zd, there are N cylindrical through holes between the two conductor plates at 1Ï, 2Ï, and NÏ respectively. According to the principle of equivalence, the equivalent magnetic field at each hole of the two conductor plates The flow pu M is located at (/ 2) pz 'dÏ, pb M is located at (, / 2) pz' dÏ, 1, 2, p N, where the subscript u indicates that the position is at the upper port of the through hole, that is, / 2 z 'd, The subscript b indicates that the position is at the port under the through hole, that is, / 2z'd. The Foldy-Lax multipath scattering equation shows that the total excitation field of the cylinder through-hole q is equal to the field source incident field plus the scattering field from all the cylinder through-holes except itself.
2.3 Modeling of multi-layer vertical vias For multi-layer vias, each single-layer via is regarded as a multi-port network. According to the multi-port network cascade theory, the scattering parameters of multi-layer vias can be obtained. Specifically, for the case of N vias in the L layer (as shown in Figure 4), it can be regarded as a cascade of L 2N port networks. After obtaining the N vias scattering parameter matrix of each layer separately The multi-port network cascade can be used to obtain the scattering parameter matrix of N vias in the L layer. Taking two layers of N vias as an example, it can be regarded as two 2N port networks. The equivalent network is shown in Figure 5. Write the "full matrix" before connection. The so-called "full matrix" refers to An S-scattering parameter matrix represents all the characteristics of simple networks before connecting them. After dividing them into ports of “unconnected†and “to be connectedâ€, the total scattering parameters after cascading L 2N ports can be further obtained That is, the scattering parameter of the L layer through hole.
3 Numerical calculation results and analysis Based on the Foldy-Lax equation and network cascading theory, this paper realizes the modeling and simulation of high-speed signal vertical vias. Compared with the NW method in the literature <6>, this method is not only suitable for porous cases , And the applicable frequency range is wider. In order to verify the effectiveness of the method in this paper, the scattering parameters of the four-layer coupled double hole are analyzed first, and the results are shown in Figure 6. In the calculation, the coaxial cable is used for feeding excitation. The inner radius of the through hole is 0.457 a mm, the outer radius is 1.524 b mm, the distance between the two holes is 12 s mm, and the distance between the two conductive plates is the thickness of the single layer of the through hole. The calculation results of document <6> are compared, and the results agree well.
In order to illustrate the influence of the surrounding scattering vias on the scattering parameters of the coupled vias, Figure 7 shows the calculation results of the scattering parameters when the four-layer double holes are distributed with 100 vias around them (as shown in Figure 8). The calculation result of the method. In this method, the Foldy-Lax multipath scattering equation is used in the calculation to consider the effect of the multipath scattering of other surrounding vias. The solid line part is the calculation result when the effect of other surrounding vias is not considered. The results show that although the reflection coefficients 1 1 uu S and transmission coefficients 1 1 bu S calculated by the two methods are not much different, the coupling coefficients between the two vias 2 1 bu S, 2 1 uu S have been greatly affected The multiple ripples are caused by the multipath scattering of randomly distributed through holes around.
4 Conclusion This paper first gives the Foldy-Lax multipath scattering equation between the through holes in a single layer, solves the excitation field coefficients of each through hole, and obtains the scattering matrix of the single layer through hole, and then applies the multi-port network cascade theory The scattering matrix of the multi-layer vertical through holes is obtained, and the modeling of the multi-layer vertical through holes is completed. Because the Foldy-Lax multipath scattering equation is used in the modeling and analysis process, compared with pure numerical methods, the calculation speed can be greatly improved when calculating the coupling problem of a large number of vertical vias. The equation itself has considered the effect of multipath scattering of the surrounding vias, so its calculation accuracy has also been improved. In modern printed circuit boards, with the increasing use of discontinuous structures such as through holes and the continuous increase of clock frequency and signal speed, the signal integrity problems caused by it will become more prominent, and the method in this paper is fast and effective Performance will help engineers analyze and design signal integrity.
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