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14 The PCB Design Magazine • February 2014 fectively analysed with quasi-static ﬁeld solvers and lines with inhomogeneous dielectric may require analysis with a full-wave solver to ac- count for the high-frequency dispersion 1, 2 . Ac- curacy of transmission line models is mostly de- ﬁned by availability of broadband dielectric and conductor roughness models. Wideband Debye (a.k.a. Djordjevic-Sarkar or Swensson-Dermer) and multi-pole Debye models 2 are examples of dielectric models suitable for accurate analysis of PCB and packaging interconnects. Expres- sion for complex permittivity of multi-pole De- bye model can be written as follows 2 : Eq. 1 Values of dielectric constant at inﬁnity e (∞) as well as pole frequencies f r n and residues De n are not known for composite dielectrics and have to be identiﬁed. The number of poles N for model suitable for analysis of interconnects up to 50 GHz should be 5-10 2 . Expression for complex permittivity of the wideband Debye model can be written as fol- lows 2 : Eq. 2 As in the case of multi-pole Debye model, a number of parameters have to be identiﬁed in Eq. 2. Values of m1 and m2 deﬁne the position of the ﬁrst and last pole in the continuous spec- trum deﬁned by the model. Those are typically set to very low and very high values outside of the frequency band of interest. Values of e (∞) and e d can be identiﬁed with only one measure- ment of dielectric constant and loss tangent 2 . f in Eq. 1 and Eq. 2 is frequency. To simulate the effect of conductor rough- ness, Huray's snowball 3 and modiﬁed Ham- merstadt 4 conductor roughness models can be effectively used. Expression for the conductor surface impedance correction coefﬁcient based on the Huray's snowball model can be written as follows 4 : Eq. 3 This model has 2 parameters: ball radius r and ratio of the number of balls to the base tile area N/Ahex. Both are not known for commonly used copper foils. Another practically useful surface imped- ance correction coefﬁcient is the modiﬁed Hammerstadt model, which can be expressed as follows 4 : Eq. 4 It has also two parameters: D or surface roughness (SR) parameter (may be associated with RMS peak to valley value) and roughness factor RF (maximal possible increase of losses due to roughness). Note that the classical Ham- merstadt model has RF=2 and just one param- eter, but this is not very useful for characterisa- tion of PCB copper 4 . d in Eq. 3 and Eq. 4 is the frequency-dependent skin depth. Manufacturers of dielectrics usually pro- vide dielectric parameters at 1–3 points in the best cases. It is not possible to construct broad- band multi-pole Debye model from just three points, to have model bandwidth from 1 MHz to 50 GHz, as typically required for 10–50 Gbps data links. Five or more points may be required with one of the points close to the highest frequency of interest 2 . In addition, all points have to be consistent and measured with the same method. Manufacturers of advanced PCB dielectric typically provide dielectric constant and loss tangent at 10 GHz or lower frequen- cies. However, those points may be acceptable for deﬁning the wideband Debye model, be- cause just one point is needed to identify the model parameters. The constructed model be- feature PCB AND PACkAGING DESIGN UP To 50 GHz continues

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