Broadband Electromagnetic and Stochastic Modelling and Signal Analysis of Multiconductor Interconnections Electrical & Computer Engineering University of Toronto Distinguished Lecture Series 2017-2018 Daniël De Zutter (Fellow IEEE) Ghent University Dept. of Information Technology Electromagnetics Group 1
Overview introduction RLGC-modelling of multiconductor lines variability analysis along the signal propagation direction analysis of statistical signals resulting from random variations in geometry, material properties, component values, linear and non-linear drivers and loads brief conclusions questions & discussion 2
Introduction How to model signal integrity? full 3D numerical tools direct access to multiport S-parameters and time-domain data holistic but time-consuming (some times too easily) believed to be accurate PlayStation 3 motherboard divide and conquer multiconductor lines , vias, connectors, packages, chips, …. model each of them with a dedicated tool derive a circuit model for each part obtain the S-parameters and time-domain data (eye-diagram, BER, crosstalk) from overall circuit representation gives more insight to the designer (optimisation) overall accuracy might be difficult to assess 3
Introduction Multiconductor Transmission Lines simplify (idealize) to a 2D problem 2D fields, charges, currents RLGC PlayStation 3 motherboard Transmission lines 3D fields, charges, voltages & currents currents 4
Multiconductor TML schematically Telegrapher’s equations (RLGC) 1 N 2 ….. reference many possibilities N+1 conductors one of which plays the role of reference conductor i : Nx1 current vector v : Nx1 voltage vector C : NxN capacitance matrix L : NxN inductance matrix G : NxN conductance matrix R : NxN resistance matrix 5
Multiconductor TML wish list number 1 broadband results (time-domain) many regions (some semi-conducting) good conductors (e.g. copper) small details reference exact current crowding and skin effect modelling + wish list number 2 variability in cross-section on-chip interconnect example: variability along propagation direction • 4 differential line pairs • semi-conducting region stochastic responses • unusual reference conductor prediction of stochastics for overall interconnects (sources, via’s, lines, ..) 6
Multiconductor TML The manufacturing process introduces variability in the geometrical and material properties but also along the signal propagation direction Photolithography: trace separation shape Impurities: permittivity, loss tangent, etc. Deterministic excitations produce stochastic responses random parameters 7
Overview introduction RLGC-modelling of multiconductor lines variability analysis along the signal propagation direction analysis of statistical signals resulting from random variations in geometry, material properties, component values, linear and non-linear drivers and loads brief conclusions questions & discussion 8
RLGC – in brief C and G can be found by solving a classical potential problem in the cross-section: cond. cond. sources/unknowns : (equivalent) boundary charges preferred method: boundary integral equation relation between total charges and voltages Q = C V diel. L and R could be found by determining the magnetic fields due to equivalent contrast currents placed in free space cond. cond. cond. cond. ? diel . diel . Suppose we find a way to replace these currents by equivalent ones on the boundaries: sources/unknowns : equivalent boundary currents preferred method: EFIE with 9
Differential surface curren t (b) (a) out out S S out in two non-magnetic media “out” & “in” we introduce a fictitious (differential) (conductor, semi-conductor, dielectric) surface current J s separated by surface S a single homogeneous medium “out” fields inside E 1 , H 1 fields inside differ: E , H fields outside E 0 , H 0 fields outside remain identical: E 0 , H 0 10
Differential Admittance Advantages modelling of the volume current crowding /skin-effect is avoided out S less unknowns are needed (volume versus surface) out homogeneous medium: simplifies Green’s function valid for all frequences losses from DC to skin effect + “internal” inductance can all be derived from J s and E tang on S Disadvantage or Challenge The sought-after J S is related to E tang through a non-local surface admittance operator in 3D How to obtain ? in 2D admittance operator similar to j z ( r ) = s e z ( r ) but no longer purely local ! 11
Differential Admittance in 2D B c S r analytically using the Dirichlet eigenfunctions of S n r’ A numerically for any S using a 2D integral equation (prof. P. Triverio) B in 3D S V r analytically using the solenoidal eigenfunctions of the volume V n r’ A see e.g. Huynen et al. AWPL, 2017, p. 1052 12
Admittance operator 45 45 26 26 B copper copper 5 m m 5 m m A 50 50 20 20 1 1 20 m m 20 m m B A 79.1 MHz - skin depth d = 7.43 m m skin depth d = 0.66 m m ( ) - 10 GHz 13
Multiconductor TML N 1 2 ….. Telegrapher’s equations (RLGC) reference wish list number 1 broadband results Final result: many regions (some semi-conducting) The 2-D per unit of length (p.u.l.) transmission line matrices R, L, G, and C , good conductors (e.g. copper) as a function of frequency small details (see ref. [5]) exact skin effect modelling 14
Examples Differential line pair e r = 3.2 s copper = 5.8 107 tan d = s / we 0 e r = 0.008 15
Examples Differential line pair L 11 = L 22 R 11 = R 22 L 12 = L 21 R 12 = R 21 16
Examples Metal Insulator Semiconductor (MIS) line s = 50S/m L DC = 422.73nH/m C DC = 481.71pF/m 17
Examples Metal Insulator Semiconductor (MIS) line @ 1GHz good dielectric good conductor 18
Examples Coated submicron signal conductor 3117 nm 500 nm 238 nm 450 nm 500 nm 450 nm copper: 1.7 mW cm chromium: 12.9 mW cm coating thickness d : 10 nm 19
Examples Coated submicron signal conductor L R inductance and resistance p.u.l as a function of frequency 20
Examples aluminum SiO 2 silicon 21
Examples Pair of coupled inverted embedded on-chip lines Discretisation for solving the RLGC-problem 22
Examples Pair of coupled inverted embedded on-chip lines: L and R results 23
Examples Pair of coupled inverted embedded on-chip lines: G and C results 24
Examples 4 differential pairs on chip interconnect + all dimensions in m m + s sig = 40MS/m + s sub = 2S/m + s dop = 0.03MS/m 25
Examples eight quasi-TM modes quasi-even quasi-odd the modal voltages V = V 0 exp(-j f ) slow wave factor: are displayed (V 0 = ) @ 10GHz mode prop. velocity v = c/SWF 26
Examples complex capacitance matrix @10GHz 27
Examples complex inductance matrix @10GHz 28
Overview introduction RLGC-modelling of multiconductor lines variability analysis along the signal propagation direction analysis of statistical signals resulting from random variations in geometry, material properties, component values, linear and non-linear drivers and loads brief conclusions questions & discussion 29
Perturbation along z What if the cross-section varies along the propagation direction? use a perturbation approach ! Quick illustration for a single line (with L & C complex – hence R & G are included) + perturbation around nominal value nominal perturbation step 1 perturbation step 2 including this second order is CRUCIAL ! 30
Example Fibre weave: differential stripline pair on top of woven fiberglass substrate differential stripline pair copper cross-section of differential stripline pair 31
Example – cont. Fibre weave - discretisation (in CAD tool) cross-section a cross-section b 32
Example – cont. Fibre weave - material properties real part of dielectric permittivity e ’ r and tan d as a function of frequency 33
Example - cont Propagation characteristics for a 10 inch line forward differential to common differential mode transmission mode conversion 34
Overview introduction RLGC-modelling of multiconductor lines variability analysis along the signal propagation direction analysis of statistical signals resulting from random variations in geometry, material properties, component values, linear and non-linear drivers and loads - PART 1: MTL brief conclusions questions & discussion 35
Monte Carlo method Interconnect designers need to perform statistical simulations for variation-aware verifications Virtually all commercial simulators rely on the Monte Carlo method Robust, easy to implement Time consuming: slow convergence 1/ N 36
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