PBLM 1 Compact Behavioural Modelling Behavioural Modelling Compact of Electromagnetic Effects of Electromagnetic Effects in On- -Chip Interconnect Chip Interconnect in On Invited presentation at Invited presentation at MACSI- -NET 2003, May 2 NET 2003, May 2- -3, 3, Zürich Zürich MACSI Peter B.L. Meijer Meijer Peter B.L. Philips Research Laboratories Philips Research Laboratories Peter.B.L.Meijer@philips philips.com .com Peter.B.L.Meijer@
PBLM 2 Contents Contents • Introduction • Introduction • Modelling “flow” “flow” • Modelling • From Maxwell’s Equations to circuit simulation • From Maxwell’s Equations to circuit simulation – Example 1: Example 1: one wire, load one wire, load modelling modelling – – Example 2: Example 2: two wires with cross two wires with cross- -talk talk – • Conclusions • Conclusions
PBLM 3 Why? Why? – Find limiting factors in Find limiting factors in – high- -speed digital circuits speed digital circuits high – Reference for validation Reference for validation – of future design rules of future design rules – Reference for validation – Reference for validation of alternative models of alternative models Source: Maly
PBLM 4 Other approaches Other approaches RLC lumped models RLC lumped models Transmission line models Transmission line models Modelling assumptions? assumptions? Modelling ROM techniques ROM techniques ... ...
PBLM 5 Maxwell Equations Modelling flow? flow? Modelling ``FDTD’’ gives V(t), I(t) Fit a linear dynamic model Our generalized formalism Post-optimize & generate syntax for simulation model Circuit Simulation
FDTD method PBLM 6 FDTD method FDTD = Finite Difference Time Domain: a method for solving the Maxwell Equations • Discretizes and and • Discretizes solves Maxwell’s solves Maxwell’s Equations in both Equations in both space and time space and time • CPU intensive! • CPU intensive! Also yields voltages and currents
PBLM 7 FDTD: CPU time = f(mesh) FDTD: CPU time = f(mesh)
PBLM 8 Using FDTD Using FDTD Mesh: 250,000 cells @ 2500 time points 1u x 1u 60 micron Al strip 80 micron SiO 2 I d e a l c o n d u c t o 5 0 r Voltage step: m i c 0.05 ps delay + r o n 0.25 ps risetime SINSQ Pstar T = 5 ps
t = 0.2 picosecond t = 0.4 picosecond t = 4.6 picosecond PBLM 9 Top view Cross section view Using FDTD Using FDTD
PBLM 10 Using FDTD Using FDTD Side view 80 micron Al strip t = 0.2 picosecond SiO 2 Perfectly conducting ground plane t = 0.4 picosecond t = 4.6 picosecond
PBLM 11 Using FDTD Using FDTD • 500 GHz reflections • 500 GHz reflections – Reflections give Reflections give – overshoot and overshoot and affect wave shape affect wave shape at source side at source side – Analyze beyond Analyze beyond – measurement options measurement options – Results sanity check: Results sanity check: – ~0.4 ps ~0.4 ps for 60 micron: for 60 micron: ~0.4 ps T = 5 ps 8 m/s ~= c / 1.5 x 10 8 m/s ~= c / � 3.9, 1.5 x 10 � 3.9, (3.9 is (3.9 is rel rel . . perm perm . . SiO SiO 2 2 ) ) Current I s (t) at source side Voltages V s (t) and V t (t)
PBLM 12 Maxwell Equations FDTD gives V(t), I(t) Circuit Simulation
PBLM 13 Linear State Space Modelling Modelling Linear State Space • Assume linear state space model: linear state space model: • Assume Matrix equations x ’(t) = ’(t) = A x A x (t) + (t) + B u B u (t) (t) Matrix equations x y (t) = (t) = C x C x (t) + (t) + D u D u (t) (t) y • Determine parameter matrices parameter matrices A, B, C, D A, B, C, D • Determine for given input vectors u (t) (t) and output and output for given input vectors u vectors y (t) (t) such that a best fit is obtained such that a best fit is obtained vectors y • MOESP/4SID class of • MOESP/4SID class of subspace identification algorithms subspace identification algorithms
PBLM 14 Linear State Space Modelling Modelling ( (MOESP/4SID) Linear State Space Behavioural modelling based on time domain data (waveforms) DeWilde, Verhaegen, Ciggaar, Meijer, Schilders Steps: Steps: • Initial order sufficiently high, e.g., sufficiently high, e.g., 200 200 • Initial order SVD plot to estimate # time constants SVD plot to estimate # time constants • Reduce order to desired value, e.g., to desired value, e.g., 10 10 • Reduce order
PBLM 15 Mapping PDE’s (Maxwell Eqs.) to equivalent high order (200) ODE system; no apparent SV clustering for selecting order Dynamic wire current modelling Using MOESP/4SID Singular value plot
PBLM 16 10-th order linear dynamic model fit (green) versus original FDTD results (red) Dynamic wire current modelling Using MOESP /4SID
PBLM 17 Maxwell Equations FDTD gives V(t), I(t) Linear state space modelling to get linear dynamic model Circuit Simulation
PBLM 18 Generalized Modelling Modelling Formalism Formalism Generalized • Map linear state space model + parameters linear state space model + parameters • Map to our neural network modelling modelling formalism formalism to our neural network Constructive and mathematically exact! Constructive and mathematically exact! • Post- -optimize optimize to deal with numerical to deal with numerical artefacts artefacts • Post of MOESP/4SID (instability & no implicit DC) of MOESP/4SID (instability & no implicit DC) • Automatically generate generate lumped linear circuit lumped linear circuit • Automatically models for Pstar Pstar, , Spectre Spectre, VHDL , VHDL- -AMS, … AMS, … models for
PBLM 19 Basic multilayer perceptron multilayer perceptron theory theory + extensions ( + extensions (Meijer Meijer 1996) 1996) Basic Weighted sum s s ik Weighted sum ik Differential equation Differential equation for neuron output y y ik for neuron output ik Inputs Outputs Feedforward neural network neural network Feedforward
Neural Networks f or Device and Circuit Modelling Neural Networks f or Device and Circuit Modelling PBLM 20 http://server506.hypermart.net/meijerpb/thesis/thesis_meijer.zip (11.5 MB) Learning ( ( = = Optimization) Optimization) Learning • Define cost function, e.g., � � (model (model - - data) data) 2 • Define cost function, e.g., 2 • Discretize and apply optimization algorithm and apply optimization algorithm* *, , • Discretize involving combinations of involving combinations of - DC, TR and AC small signal analysis DC, TR and AC small signal analysis - - DC, TR and AC sensitivity (for gradients) DC, TR and AC sensitivity (for gradients) - *Conjugate gradient, BFGS, … Conjugate gradient, BFGS, … * • Risks: slow convergence, local minima • Risks: slow convergence, local minima
PBLM 21 Applying Generalized Formalism Applying Generalized Formalism Fix MOESP/4SID artefacts: Post-optimize ensure stable model & fit with DC initial state
PBLM 22 Maxwell Equations FDTD gives V(t), I(t) Linear state space modelling to get linear dynamic model Our generalized formalism Post-optimize & generate syntax for simulation model (lumped linear circuit model) Circuit Simulation
PBLM 23 FDTD results and FDTD results and Neureka Neureka/ /Pstar Pstar NN model results NN model results Circuit simulation results vs vs FDTD FDTD Circuit simulation results
PBLM 24 Verify model generalization Verify model generalization • Linear model: modelling modelling for one signal with all for one signal with all • Linear model: (relevant) frequencies should suffice - - in theory! in theory! (relevant) frequencies should suffice • Verify: define a define a different stimulus different stimulus and check if and check if • Verify: the FDTD simulation still matches results for the the FDTD simulation still matches results for the unchanged circuit model simulation circuit model simulation unchanged If so, that will confirm that the circuit model If so, that will confirm that the circuit model indeed applies to all stimuli all stimuli - - and not just the and not just the indeed applies to one(s) used during modelling modelling one(s) used during
PBLM 25 Verify model generalization [1] Verify model generalization [1] • New stimulus stimulus • New STEP with STEP with slope /= 10, slope /= 10, applied to applied to – FDTD FDTD – simulation simulation – Unchanged Unchanged – circuit circuit Circuit simulation simulation simulation model versus FDTD model model Excellent fit!
PBLM 26 Verify model generalization [2] Verify model generalization [2] • New stimulus stimulus • New GAUSSIAN GAUSSIAN applied to applied to Circuit simulation model versus FDTD – FDTD FDTD – simulation simulation – Unchanged Unchanged – circuit circuit simulation simulation model model Excellent fit!
PBLM 27 Single Wire Modelling Modelling Single Wire FDTD simulation of FDTD simulation of Lumped linear dynamic Lumped linear dynamic Maxwell Equations Maxwell Equations circuit simulation model circuit simulation model in space and time in space and time Complex wire load modelling # Model Eqs. 10 1 - 10 2 # Model Eqs. 10 5 - 10 6 Orders of magnitude gain in simulation speed while preserving detailed (parasitic) effects
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