Transcript fasterix

Model Order Reduction for
EM modeling of IC's
Maria Ugryumova
CASA day, November 2007
Contents
Introduction
•
Simulation of EM behaviour
•
Boundary Value Problem, Kirchhoff’s equations
•
Reduced Order Modelling for EM Simulations
•
The Approach of EM Simulation used in FASTERIX
•
Transient Analysis
•
Simulation for FULL Model in Time Domain
Super Node Algorithm
•
Details of Super Node Algorithm
•
Observations
•
Examples: Transmission line model, Lowpass filter
•
Conclusions on Super Node Algorithm
•
Passivity
•
Future work
2
Simulation of EM behaviour
Printed Circuit Board
Equivalent Circuit
Model
Currents through Conductor
radiated EM fields
Simulator
3
Simulation of EM behaviour
Printed Circuit Board
Equivalent Circuit
Model
Currents through Conductor
radiated EM fields
Simulator
3
Simulation of EM behaviour
Printed Circuit Board
Equivalent Circuit
Model
Currents through Conductor
radiated EM fields
Simulator
3
Simulation of EM behaviour
Printed Circuit Board
Equivalent Circuit
Model
Currents through Conductor,
radiated EM fields
Simulator
3
Boundary Value Problem
J

   i  G A Jdx '  E0

  J  i  0

   G dx'  E0


BC:
 ( x)  V fixed x  V ,
J  n  0, x  ,
  L2 () - charge density
  H 1 () - scalar potential
J  H div () - current density
E0 - irradiatio n
 - permeabili ty
 - permittivi ty
 - conductivi ty
Discretisation
Set of quadrilateral elements:  j , j  1,...Nelem
Set of edges (excluding element edges in boundary):
4
 l , l  1,...Nedge
Kirchhoff’s equations
(R  j L) I  PV  0

-PI +jCV  J
R
C
L
P
(n  n) - resistance
(m  m) - capacitance
(m  m) - inductance
(n  m) - incidence
matrices from
FASTERIX
• FASTERIX – to simulate PCB’s
• Matrix coefficients are integrals. They are frequency independent
• Linear set of (Nedge + Nelem) equations
• Solved for I – currents over branches and V – potentials at the nodes
• J – collects currents flowing into interconnection system
5
Approach
Simulation of
PCB’s
Model Order
Reduction
FASTERIX
6
Reduced Order Modelling for EM Simulations
Model Order Reduction
•
Preservation of passivity
•
PRIMA, Laguerre SVD
FASTERIX - layout simulation tool
Super Node Algorithm (SNA)
SNA delivers models based on max applied frequency
Good results in Frequency Domain
SNA produces not passive models
7
Reduced Order Modelling for EM Simulations
Model Order Reduction
•
Preservation of passivity
•
PRIMA, Laguerre SVD
FASTERIX - layout simulation tool for EM effects
•
Super Node Algorithm (SNA)
•
SNA delivers models based on max applied frequency
•
Good results in Frequency Domain
•
SNA produces not passive models
7
The Approach of EM Simulation used in FASTERIX
1. Subdivision PCB into quadrilateral elements i  
2. Equivalent Circuit (EQCT)
•
Each finite element i corresponds to i-th node;
•
Each pair of neighbour nodes is connected with RL-branch;
•
Such large model is inefficient for simulator;
3. Reduced Equivalent Circuit
• Built on accessible nodes + some internal nodes from EQCT;
• The higher user-defined frequency, the more super nodes;
• Each pair of super nodes has RLGC-branch.
4. Simulation for Transient Analysis
8
Transient Analysis
PSTAR
simulator
V(OUT)
Voltage in the node Rout is measured depending on time.
Rise time << Time of propagation
9
Simulation for FULL Model in Time Domain
In order to get solution for full model to compare results with
Input: Pulse, rise time = 100ps
Output: Voltage on the resistor Rout
10
Contents
Introduction
•
Simulation of EM behaviour
•
Boundary Value Problem, Kirchhoff’s equations
•
Reduced Order Modelling for EM Simulations
•
The Approach of EM Simulation used in FASTERIX
•
Transient Analysis
•
Simulation for FULL Model in Time Domain
Super Node Algorithm
•
Details of Super Node Algorithm
•
Observations
•
Examples: Transmission line model, Lowpass filter
•
Conclusions on Super Node Algorithm
•
Passivity
•
Future work
11
Super Node Algorithm [Cloux, Maas, Wachters]
• Geometrical details are small compared with the wavelength of operation
• The subdivision of the set of nodes: N=N U N’
V 
C
V   N '  , given VN   mn , C   N ' N '
 VN 
 CNN '
CN ' N 
 , P   PN '
CNN 
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PN  ,
 0 
J  
 JN 
Super Node Algorithm [Cloux, Maas, Wachters]
• Geometrical details are small compared with the wavelength of operation
• The subdivision of the set of nodes: N=N U N’
V 
C
V   N '  , given VN   mn , C   N ' N '
 VN 
 CNN '
(R  j L) I  PV  0

-PI +jCV  J
CN ' N 
 , P   PN '
CNN 
PN  ,
( R  j L) I  PN 'VN '  PNVN
 T
 PN ' I  jCN ' N 'VN '  jCN ' NVN
 0 
J  
 JN 
Depends
on frequency 
J N  PNT I  jCNN 'VN '  jCNNVN
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Details of Super Node Algorithm
• Geometrical details are small compared with the wavelength of operation
• The subdivision of the set of nodes: N=N U N’
V 
C
V   N '  , given VN   mn , C   N ' N '
 VN 
 CNN '
(R  j L) I  PV  0

-PI +jCV  J
jCmn
j L jk
Z 01O( jk0 h)
Z 0O ( jk0 h)
Rkl  Z sO(1)
CN ' N 
 , P   PN '
CNN 
PN  ,
( R  j L) I  PN 'VN '  PNVN
 T
 PN ' I  jCN ' N 'VN '  jCN ' NVN
 0 
J  
 JN 
Depends
on frequency 
J N  PNT I  jCNN 'VN '  jCNNVN
VNn'  V0n  V1n
I n  I 0n  I1n
where k0 is wave number
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( R  j L) I 0n  PN 'V0n  PNVNn
 T n
 PN ' I 0  0
( R  j L) I1n  PN 'V1n  0
 T n
n
n
 PN ' I1  jCN ' N 'V0  jCN ' NVN
Details of Super Node Algorithm
• Admittance matrix:
J Nn  PNT ( I 0n  I1n )  sCNN 'V0n  sCNNVNn , s  i
Solving Kirchhoff's equation independent on frequency
13
Details of Super Node Algorithm
• Admittance matrix:
J Nn  PNT ( I 0n  I1n )  sCNN 'V0n  sCNNVNn , s  i
Solving Kirchhoff's equation independent on frequency
• For high frequency range: ( R  sL)
I 0n  s 1 I 00n  s 2 I 01n
V0n  V00n  s 1V01n
I1n  sI10n  I11n
V1n  s 2V10n  sV11n
sL
 LI 00n  PN 'V00n  PNVNn
 T n
 PN ' I 00  0
 LI 01n  PN 'V01n   RI 00n
 T n
 PN ' I 01  0
( I kl ,Vkl ), k , l =1,4 - frequency independ. quantities
13
 LI10n  PN 'V10n  0
 T n
 PN ' I10  C N ' N 'V00  C N ' NVN
 LI11n  PN 'V11n   RI10n
 T n
 PN ' I11  C N ' N 'V01
Details of Super Node Algorithm
• Admittance matrix:
J Nn  PNT ( I 0n  I1n )  sCNN 'V0n  sCNNVNn , s  i
Solving Kirchhoff's equation independent on frequency
• For high frequency range: ( R  sL)
I 0n  s 1 I 00n  s 2 I 01n
V0n  V00n  s 1V01n
I1n  sI10n  I11n
V1n  s 2V10n  sV11n
sL
 LI 00n  PN 'V00n  PNVNn
 T n
 PN ' I 00  0
 LI 01n  PN 'V01n   RI 00n
 T n
 PN ' I 01  0
 LI10n  PN 'V10n  0
 T n
 PN ' I10  C N ' N 'V00  C N ' NVN
 LI11n  PN 'V11n   RI10n
 T n
 PN ' I11  C N ' N 'V01
( I kl ,Vkl ), k , l =1,4 - frequency independ. quantities
• Admittance matrix: J n  s 2Y  s 1Y  Y  sY
N
R
L
G
C
Branch of Reduced Equivalent
Circuit
13
Observations
• Problems of modeling in Time Domain
V(Rout)
Lowpass filter
Max freq = 10GHz
257 elements
98 supernodes
Reduced model by SNA
Full model by SNA
• Original BEM discretisation leads to passive systems
• Super Node Algorithm is based on physical principals
14
Observations
• Problems of modeling in Time Domain
V(Rout)
Lowpass filter
Max freq = 10GHz
257 elements
98 supernodes
Reduced model by SNA
Full model by SNA
Modified SNA
• Original BEM discretisation leads to passive systems
• Super Node Algorithm is based on physical principals
14
Increasing the number
of
super nodes?
Lowpass filter
Max frequency = 7 GHz
V(Rout)
FASTERIX: 227 Elements,
40 Super Nodes
Experiment
Fine mesh: 2162 Elements
full unreduced model
40 nodes
85 nodes
T
16
Transmission Line Model
Max frequency = 3 GHz
Time delay ~ 1.3 ns.
FASTERIX: 160 Elements,
100 Super Nodes
V(Rout)
Experiment
Fine mesh: 1550 Elements
50 nodes
120 nodes
full unreduced model
• Increasing of super nodes
on the fine mesh gives more
accurate results but has an
upper limit.
T
15
• Increasing of super nodes
does not give “right” properties
of admittance matrices.
Conclusions on Super Node Algorithm
• Super node algorithm is motivated by physical and electronic insight.
It is worth to modify it;
• Decreasing the distance between super nodes does not ensure passivity of the reduced
model;
• Increasing of super nodes on the fine mesh gives more accurate results but has
an upper limit for simulator;
• Necessity of detailed analysis of properties of projection matrix P due to guaranty
passivity of the models.
17
Passivity
• Incapable of generating energy;
• The transfer function H(s) of a passive system is positive real, that is,
H(s) is analytic for all s with Re(s) > 0
H ( s )  H *( s )  Re( H ( s )) for s  C
H ( s)  H *( s)  Re( H ( s))  0 for s : Re(s)>0
18
Passivity
• Incapable of generating energy;
• The transfer function H(s) of a passive system is positive real that is
H(s) is analytic for all s with Re(s) > 0
H ( s )  H *( s )  Re( H ( s )) for s  C
H ( s)  H *( s)  Re( H ( s))  0 for s : Re(s)>0
Before reduction
(G  sC ) x  Bv ext

T
i  L x
Eigenvalues of G have non-negative real part, C is symmetric positive semi-definite;
After reduction by SNA
G, C - indefinite; have the same number of positive and negative eigenvalues;
G 1C will have positive eigenvalues;
After projection: ( P*GP), ( P*CP) can have diff. number of pos. and neg. eigenvalues.
We need to define properties of P to have all positive eigenvalues.
18
Future work
• Investigation of matrix properties and eigenvalues when increasing the number of
super nodes;
• Deriving a criterion for choosing super nodes that guarantees passivity;
• Implementation in FASTERIX and comparison with MOR algorithms;
• Making start with EM on IC problem for SiP.
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Thank you for attention