Parasitic Extraction Luca Daniel University of California, Berkeley

Parasitic Extraction Luca Daniel University of California, Berkeley
Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University of California, Berkeley Joel Phillips, Cadence Berkeley Labs Jacob White, Massachusetts Instit. of Technology

Conventional Design Flow
Funct. Spec Logic Synth. Gate-level Net. RTL Layout Floorplanning Place & Route Front-end Back-end Behav. Simul. Gate-Lev. Sim. Stat. Wire Model Parasitic Extrac.

Layout parasitics Wires are not ideal. Parasitics: Why do we care?
Resistance Capacitance Inductance Why do we care? Impact on delay noise energy consumption power distribution Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic

Parasitic Extraction thousands of wires e.g. critical path
e.g. gnd/vdd grid Parasitic Extraction identify some ports produce equivalent circuit that models response of wires at those ports tens of circuit elements for gate level spice simulation

Parasitic Extraction (the two steps)
Electromagnetic Analysis million of elements thin volume filaments with constant current small surface panels with constant charge Model Order Reduction tens of elements

Overview Setups of Parasitic Extraction Problems
Capacitance Extraction (electrostatic) RL Extraction (MQS) Combined RLC Extraction (EMQS) Electromagnetic Interference Analysis (fullwave) Electromagnetic solvers classification (time vs. frequency, differential vs. integral) integral equation solvers in detail basis functions residual minimization (collocation and Galerkin) linear system solution fast matrix-vector products Example: EMQS solution Conclusions

Capacitive Extraction Example: Intel 0.25 micron Process
5 metal layers Ti/Al - Cu/Ti/TiN Polysilicon dielectric. Taken from “Digital Integrated Circuits”, 2nd Edition, Rabaey, Chandrakasan, Nikolic Consider only electric field (capacitive) coupling

Capacitive Extraction Why? E.g. Analysis of Delay of Critical Path

Capacitance Extraction Why do we need it?
2 Example: to produce RC tree network for elmore delay analysis Example: to produce RC tree network for capacitive cross-talk analysis R2 C2 R1 4 1 s R4 C4 R3 C1 3 Ci Ri C3 i

Capacitance Extraction Problem Formulation
Given a collection of N conductors (of any shape and dimension) Calculate the coupling capacitance matrix C

Capacitance Extraction Solution Procedure
For i = 1 to N, apply one volt to conductor i and ground all the others solve the electrostatic problem and find the resulting vector of charges on all conductors that is the i-th column of the conductance matrix

Inductance and Resistance Extraction Example: IC package
Picture Thanks to Coventor wire bonding lead frames IC package

Inductance and Resistance Extraction Where do we need to account for inductance?
chip to package and package to board connections are highly inductive inductance can create Ldi/dt noise on the gnd/vdd network inductance can limit communication bandwidth inductive coupling between leads or pins can introduce noise pins or solder balls from package to PCB wire bonding and lead frames or solder balls from IC to package IC PCB package on-board decoupling capacitors on-package decoupling capacitors

Inductance and Resistance Extraction Why also resistance
Inductance and Resistance Extraction Why also resistance? Skin and Proximity effects Simple Example proximity effect: opposite currents in nearby conductors attract each other skin effect: high frequency currents crowd toward the surface of conductors

Inductance and Resistance Extraction Skin and Proximity effects (cont
Why do we care? Skin and proximity effects change interconnect resistance and inductance hence they affect performance (propagation delay) and noise (magnetic coupling) When do we care? frequency is high enough that wire width OR thickness are less than two “skin-depths” e.g. on PCB at and above 100MHz e.g. on packages at above 1GHz e.g. on-chip at and above 10GHz note. clock at 3GHz has significant harmonics at 10GHz!!

Inductance and Resistance Extraction Problem Formulation
Given a collection of interconnected N wires of any shape and dimension Identify the M input ports Picture by M. Chou Calculate the MxM resistance and the inductance matrices for the ports, that is the real and immaginary part of the impedance matrix

Inductance and Resistance Extraction Solution Procedure
Typically instead of calculating impendance we calculate the admittance matrix. For each pair of input terminals, apply a unit voltage source and solve magneto quasit-static problem (MQS) to calculate all terminal currents that is one column of the admittance matrix [R+jwL]-1

Combined RLC Extraction Example: current distributions on powergrid
input terminals

Combined RLC Extraction Example: analysis of resonances on powergrid
* 3 proximity templates per cross-section - 20 non-uniform thin filaments per cross-section

Combined RLC Extraction Extraction Example: analysis of substrate coupling

Combined RLC Extraction Example: resonance of RF microinductors
At frequency of operation the current flows in the spiral and creates magnetic energy storage (it works as an inductor: GOOD) But for higher frequencies the impedance of the parasitic capacitors is lower and current prefers to “jump” from wire to wire as displacement currents (it works as a capacitor: BAD) Picture thanks to Univ. of Pisa

Combined RLC Extraction Problem Formulation
Given a collection of interconnected N wires of any shape and dimension Identify the M input ports Picture by M. Chou Calculate the MxM IMPEDANCE matrix for the ports, that is the real and immaginary part of the impedance matrix

Combined RLC Extraction Solution Procedure
Same as RL extraction. Typically calculate admittance matrix For each pair of input terminals, apply a unit voltage source and solve electro-magneto quasit-static problem (EMQS) to calculate all terminal currents that is one column of the admittance matrix [R+jwL]-1

The Electromagnetic Interference (EMI) Problem description
Electronic circuits produce and are subject to Electromagnetic Interference (EMI). in particular when wavelengths ~ wire lengths EMI is a problem because it can severely and randomly affect analog and digital circuit functionality!!! PCB IC

EMI analysis EMI at board, package and IC level
Traces on PCB can pick up EMI and transmit it to IC’s IC’s can produce high frequency conducted emissions that can radiate from PCB’s IC’s themselves can directly produce radiated emissions high-frequency current loops Vdd-decap-gnd on package or inside IC’s. high-frequency current loops inside IC (near future) IC radiation amplified by heat sinks! IC IC PCB IC PCB

EMI a problem for ICs design?
So far: dimensions too small and wavelengths too large Trend: larger chip dies and higher frequencies Today’s PCB: clocks ~ 300MHz harmonics ~ 3GHz wavelengths ~ 10cm dimensions ~ 10cm this gives resonances on PCB today, hence it might on IC tomorrow! Future’s IC: clocks ~ 3GHz harmonics ~ 30GHz wavelengths ~ 1cm dimensions ~ 1cm

EMI analysis Solution Procedure
Typically, EMI analysis is a two-step process: 1) determine accurate current distributions on conductors 2) calculate radiated fields from the current distributions E

Need for full-board analysis
Interconnect impedances depend on complicated return paths. Unbalanced currents generate most of the interference. Hence need FULL-BOARD analysis

Need for full-wave analysis
Circuit dementions are not negligible compared to wavelength coupling NOT instantaneus, speed of light creates retardation Need to solve FULLWAVE equations (same as for RLC extraction plus wave term)

Capacitance Extraction (electrostatic) RL Extraction (MQS) Combined RLC Extraction (EMQS) Electromagnetic Interference Analysis (fullwave) Electromagnetic field solvers classification (time vs. frequency, differential vs. integral) integral equation solvers in detail basis functions residual minimization (collocation and Galerkin) linear system solution fast matrix-vector products Example: EMQS solution Conclusions

Example: the most intuitive FDTD in one dimension using Forward Euler (easier to explain)
Maxwell differential equations: In one dimension: Using forward Euler:

Example: 1D-FDTD with Forward Euler (cont.)
Iteration formulas: t n+1 n m x m+1

Time-domain vs. frequency domain methods
Time-domain methods Frequency-domain methods can handle non-linearities problems with non-linearities run a long simulation exciting solve for specific frequency all significant modes and then points of interest take an FFT can produce insightful can exploit new techniques animations for model order reduction

Differential vs. Integral methods
Differential methods Integral Methods discretize entire domain discretize only “active” regions create huge but sparse create small but dense linear systems linear systems good for inhomogeneous problems with materials inhomogeneous materials problems with open good for open boundary boundary conditions conditions

Electromagnetic Solvers Classification
Differential Integral Methods Methods Time-domain FDTD PEEC Methods Finite Difference Partial Element Time Domain Equivalent Circuits Frequency-domain FEM MoM, PEEC Methods Finite Element Method of Moments Method

Maxwell Differential Equations
Maxwell Differential Equations can be written in terms of the electric scalar potential and the magnetic vector potential

Maxwell equations in integral form Mixed potential Integral Equation (MPIE)
charge-voltage relation resistive effect magnetic coupling current and charge conservation

Full-wave (for EMI) vs. quasi-static EMQS (for RLC extraction)
charge-voltage relation resistive effect magnetic coupling QUASI-STATIC QUASI-STATIC current and charge conservation

EMQS (for RLC extraction) vs. MQS (for RL extraction)
resistive effect magnetic coupling MQS MAGNETO QUASI-STATIC charge-voltage relation current and charge conservation

EMQS (for RLC extraction) vs
EMQS (for RLC extraction) vs. electro-static (for capacitance extraction) ELECTRO-STATIC resistive effect magnetic coupling charge-voltage relation ELECTRO-STATIC current and charge conservation

Basis Functions Basis for vector space
Basis functions for functional vector space Examples exponentials cos sin pieacewise constant pieacewise linear

Discretize Surface into Panels
Piecewise Constant Basis Functions. E.g. Capacitance Extraction Electrostatic problem Integral Equation: Discretize Surface into Panels Panel j

Residual Definition and Minimization

Residual Minimization using Test Functions
Note: Weighted Residual = Galerkin when using piecewise constant basis functions

Collocation

Put collocation points at
Collocation on Piecewise constant basis functions (Centroid Collocation) Put collocation points at panel centroids Collocation point

Centroid Collocation generates a nonsymmetric matrix

Basis Function Approach Calculating Matrix Elements
z Collocation point y x Panel j One point quadrature Approximation Four point quadrature Approximation

Basis Function Approach Calculating “Self-Term”
z Collocation point y x Panel i One point quadrature Approximation

Basis Function Approach Calculating “Self-Term” Tricks of the trade
z Collocation point y x Panel i Disk of radius R surrounding collocation point Integrate in two pieces Disk Integral has singularity but has analytic formula

1) If panel is a flat polygon, analytical formulas exist
Basis Function Approach Calculating “Self-Term” Other Tricks of the trade z Collocation point y x Panel i 1) If panel is a flat polygon, analytical formulas exist 2) Curve panels can be handled with projection

Galerkin A is symmetric

Galerkin with Piecewise constant bases
test panel i source panel j

Background Krylov subspace iterative methods
Problem: solve Ax = b (where A is large and dense) guess x0 REPEAT how good was my guess? Calculate residue ri = b-Axi find next guess xi+1 from the space x0+span{r0,Ar0,A2r0,…,Air0} which minimizes the next residue UNTIL ||residue|| < desired accuracy Advantages over gaussian elimination: 1) get a great control on accuracy (can stop and save computation when desired accuracy is achieved) 2) only need a matrix vector product Ax per iteration: O(N2)

Background Solving the large DENSE linear system (cont)
Solving the dense large linear system obtained from the Integral Equation method:

Precorrected-FFT (e.g. capacitance extraction)
Problem: solve iteratively Pq=Φ At each iteration evaluate matrix- vector products Pq: (1) Project charges into a 3D grid (2) calculate grid potentials it is a convolution use FFT. (3) Interpolate potentials from grid but calculate close interactions directly (pre-correction step) (1) (3) (2) Picture by J. Phillips O(n2) O(n log n)

Frequency domain integral equation method EMQS Mixed Potential Integral Equation
charge-voltage relation resistive effect magnetic coupling current and charge conservation

EMQS Basis functions + Galerkin
Standard numerical procedure: discretize volumes and charges into piecewise constant basis functions. small surface panels with constant charge thin volume filaments with constant current Substitute and use Collocation or Galerkin to get branch equations

Mesh Analysis [Kamon et al. Trans Packaging98]
Branch equations ú û ù ê ë é = + c V q I P L j R f w Imposing current conservation with mesh analysis

EMQS, mesh analysis, Preconditioners
A Preconditioner must be a good approximation of the inverse of the matrix easy to calculate Jacobi Preconditioner M is sparse A better choice

GMRES convergence with different preconditioners

Parasitic Extraction (the two steps)
Electromagnetic Analysis (Tuesday) million of elements thin volume filaments with constant current small surface panels with constant charge Model Order Reduction (Today) tens of elements

Parasitic Extraction thousands of wires e.g. critical path
e.g. gnd/vdd grid Parasitic Extraction identify some ports produce equivalent circuit that models response of wires at those ports tens of circuit elements for gate level spice simulation

Conventional Design Flow
Funct. Spec Logic Synth. Gate-level Net. RTL Layout Floorplanning Place & Route Front-end Back-end Behav. Simul. Gate-Lev. Sim. Stat. Wire Model Parasitic Extrac.

Parasitic Extraction Luca Daniel University of California, Berkeley

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