Digital Wave Formulation of Quasi-Static

Partial Element Equivalent Circuit Method

Piero Belforte, Luigi Lombardi,

Daniele Romano, Giulio Antonini

piero.belforte@gmail.com , luigilombardi89@gmail.com,

daniele.romano.vis@gmail.com, giulio.antonini@univaq.it,

SPI 2016

Torino, May 10, 2016

Summary

o Basic PEEC theory

o Digital Wave Networks

o Digital Wave PEEC Network

o Solution algorithm

o Digital Wave Simulator

o Numerical results

o Conclusions 2

3

PEEC-based modeling

PEEC

modeling Materials modeling

Frequency and

time domain analysis

Integration with

circuit solvers

Wideband models:

from DC to daylight

Efficient solvers

Basic PEEC Theory

Invented by A. Ruehli (IBM) through the concepts of

partial inductance (1972)

partial capacitance (1973)

integrating them into the same formulation (1974)

4

PEEC time domain MNA solver

Quasi-static PEEC time domain solver (ODE)

T

p

dx tC Gx t Bu t

dt

y t L x t

5

Typically a large equivalent

circuit is generated.

It can be easily mapped into

Spice-like environments.

Digital wave network (DWN)

6

• PEEC model analysis are tipically performed in

current and voltage (or sometimes charge) variables.

• A possible alternative approach can use incident and

reflected voltage wave variables.

DWN is not just a change of variables !!!

7

Continuous to discrete time transform

Bilinear transform Trapezoidal rule

Inductance

The computation of the reflected wave for the next time

step is completely explicit !!!

8

Continuous to discrete time transform

Capacitance:

Resistance:

Analogously ...

• Constitutive equations are explicit!!

• … nevertheless Kirchoff laws still enforce implicit

equations…

• ... we can get a more explicit scheme by introducing

delays on coupling modeling.

Adaptors

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In order to transfer signal (voltages or currents) between circuital

elements we use series or parallel connection.

In the wave domain the equivalent concept is represented by adaptors.

Reflection-Free Port Series Adaptors

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Reflection-Free Port

Reflection-Free Port Parallel Adaptors

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Reflection-Free Port

Marx model for inductive coupling

12

Multiple coupling requires the

computation of reluctances

(acceleration techniques).

Mutual inductors are

delayed by one time step.

Link model for inductive coupling

13

Equivalent digital network for the inductive

coupling Pi model

Link model for the inductive coupling Pi

model

RLC PEEC 2-cells Model

14

PEEC 2-cells Model with VCVS

15

Equivalent digital network

16

ASc adaptors connect the capacitive

portion of the PEEC model

ASs and ASL connect source and load

respectively

NL parallel adaptors allow us to represent

the Marx model

ASRL adaptors build the RL branches

equivalent

17

Root of the digital network

The N parallel adaptors

will become the nodes of the Root

The AS series adaptors

will become the branches of the Root

We have a loop that prevents

the explicit resolution of the

innermost part of the digital

network.

Root circuital representation

18

The root of the digital network can be

solved by Nodal Analysis of an

equivalent electrical circuit.

19

Incident waves

update on leaves

Incident waves

propagation (leaves to

root)

Root

Reflected waves

propagation (root to leaves)

Circuital elements to

wave elements

Port impedances computation

Adaptor scattering parameters

computation

(Reflection-Free Port)

Circuit-oriented tool, alternative to MNA solvers.

A semiexplicit scheme has been utilized but

several other schemes can be used, some of them

being fully explicit.

Digital Wave Simulator (DWS) • Development started at CSELT Labs (Turin) in 1974 by P. Belforte & G.

Guaschino for design of high-speed digital systems

• From 1986 to 2001 at HDT (Turin) as general purpose Spice-like simulator

(SPRINT). SI/PI/EMC applications included PRESTO (post-layout),

EMIR (emissions) and THRIS in cooperation with CSELT (Qualification

tool)

• In 1998 at HDT first DWS-PEEC application (3D_PEEC)

• From 2001 to present as DWS including Multi-gigabit applications as

HiSAFE for Cisco Systems (P. Belforte)

• From 2012 also as Spicy SWAN cloud-based app

• From Feb. 2016 new PEEC-DWS developments in a cooperation driven by

P.Belforte and G.Antonini.

20

DWS early applications (1975)

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CSELT LABS: ETA (Easy Transient Analysis) application to the design of a .5Gbps

high-speed Multichip Module. A TDR-based wideband measurement system was

included in the Digital Wave modeling & simulation environment.

DWS main features

• Conversion of a Spice-like netlist into a Digital Network

equivalent including circuital elements and nodes as

scattering blocks exchanging waves at their ports.

• DSP oriented solution apart from the root. DFLs solved by

port-matching calculation scheduling

• Wideband SI/PI/EMC applications

• Complementary to Spice

Detailed documents available at

https://www.researchgate.net/profile/Piero_Belforte

22

Numerical results

23

L = 116 mm 50, 100, 2x100 and 2X200 cells

Microstrip

Intel Quad-Core i7-2630QM 2.00 GHz CPU

(100 ns window)

23

5-conductor MTL

25

1550 lines netlist

Simulation time (10fs DWS tstep,

ngspice tmax):

Ngspice 3000 sec

DWS 79 sec

Speed-up 38x

Power divider

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35500 lines netlist

Simulation time (500fs):

Ngspice ̴ 3450 sec

DWS 5,5 sec

Speed-up 627x

Conclusions

27

• A Digital Wave (DW) model of quasi-static PEEC circuits has

been proposed.

• A proper scheduling of calculations has been used.

• Significant speed-ups (up to 627x) have been experienced

replacing MNA Spice-like solvers with DWS.

• DWS speed-up increases with PEEC model complexity.

• A semi-explicit scheme has been tested so far but...

• ...at least 7 more implementations including fully explicit schemes

with different stability properties and performances are possible.

They are under investigation.

Future work

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• Inclusion of physical delays leading also to a fully

explicit scheme.

• Stability and passivity analysis of delayed wave digital

network.

• Developement of a in-house Digital Wave PEEC Solver

exploiting the features of all the possible topologies.

• Inclusion of skin-effect and dielectric losses.

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Thank you for your attention !