A Hierarchical Approach to the Stochastic Analysis of ... · 23 rd IEEE Workshop on Signal and...
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23rd IEEE Workshop on Signal and Power Integrity (SPI 2019), June 18-21, Chambéry, France
A Hierarchical Approach to the
Stochastic Analysis of Transmission Linesvia Polynomial Chaos
P. Manfredi and R. Trinchero
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Manufacturing tolerances
Predicted and measured responses can differ because of manufacturing
variability, regardless of model accuracy
simulation
measurement
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Uncertain parameters are collectively denoted as �
Stochastic responses are modeled as expansions of polynomials
orthonormal to the distribution of � [1]
Polynomial chaos expansion (PCE)
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� �, � ≈ � �� � ��(�)�
�
orthonormal polynomialsdeterministic coefficients
uncertainparameters
UncertaintyQuantification
Calculation of PCE coefficients typically much faster than Monte Carlo
deterministic input stochastic
output
[1] P. Manfredi, D. Vande Ginste, I. S. Stievano, D. De Zutter, and F.G. Canavero, “Stochastic transmission line analysis via polynomial chaos methods: an overview,” IEEE Electromagn. Compat. Mag. (2017).
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Simulation flow for lumped circuits
Physical level Component level Electrical output level
L
C
Variability is usually modeled at component level
voltagescurrents
Geometry, materialsDistribution:standard (e.g., Gaussian)
Component valuesDistribution: standard ornon-standard (but still independent)
PCE w.r.t.component values
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23rd IEEE Workshop on Signal and Power Integrity (SPI 2019), June 18-21, Chambéry, France
terminalvoltages / currents
per-unit-lengthRLGC
parameters
terminalvoltages / currents
per-unit-length RLGC
parameters
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Simulation flow for distributed transmission lines
Physical (low) level Component (mid) level Electrical (high) output level
Variability is modeled at physical level because
per-unit-length parameters are NOT independent!
variability
Modeling at component
level possible?
PCE w.r.t.physical params
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Step #0: identification of random variables
A hierarchical approach is implemented, in which the new random
variables are (mid-level) entries of per-unit-length matrices [2]
A different variable is assigned to each distinct entry:
Other properties/assumptions (e.g., PEC planes, shields, homogeneity)
may lead to further reduction in the number of mid-level parameters �
=��� ��� ������ ��� ��� ��� ��� ���
=�� �� ���� �� �� �� �� ��
reciprocity!
� =��� ��� ������ ��� ��� ��� ��� ���
=�� �� ���� ��� ��� �� ��� ���
� = ��, ��, …
[2] P. Manfredi, “A hierarchical approach to dimensionality reduction and nonparametric problems in the polynomial chaos simulation of transmission lines,” IEEE Trans. Electromagn. Compat. (early access)
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1. Dimensionality reduction when per-unit-length (mid-level) parameters are
fewer than physical (low-level) parameters
2. Can deal with nonparametric problems, for which low-level parameters
cannot be explicitly defined
3. Can achieve higher accuracy for a given expansion order
1:1 correspondence between physical and per-unit-length params
low level mid level high level
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Advantages
Cardinality: � < �
� � ≈ � ���� ��
�� � ≈ � �!��!�(�)
�
�� = �per-unit-length
parametersclassical PCE hierarchical PCE
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Step #1: mixture of Gaussians (MoG) fit
Empirical distribution of per-unit-length parameters (dependent entries �)
is fitted using a mixture of Gaussians (MoG)
A MoG is a weighed combination of correlated Gaussian distributions
" � = � #$%&�
� �&'( )*(+,(�&'()
det(21*$)�2
$3�
weights means
covariance matrices
analyticalmodel
1D
2D4 = 2
4 = 3
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Step #2: calculation of suitable basis functions
Classical PCE uses standard polynomials (available a priori)
Suitable basis function to be computed for correlated MoG distribution
This is achieved through Gram-Schmidt orthogonalization [3]
[3] C. Cui and Z. Zhang, “Stochastic collocation with non-Gaussian correlated process variations: theory, algorithms and applications,” IEEE Trans. Compon. Packag. Manuf. Technol. (early access).
Linearly independent monomials Ψ�
expectations = scalar coefficients(computed analytically or numerically)
previously-computed polynomials(iterative procedure)
(normalization)
�7� = Ψ� − � E Ψ��!: �!:�&�
:3��!� = �7�
E �7���
(orthogonalization)
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Step #3: stochastic Galerkin method
Voltage and current PCE coefficients are computed via stochastic
Galerkin method [1]
To obtain transient results, the augmented transmission line can be:
Simulated directly into SPICE (lossless lines)
Solved in frequency domain and results post-processed with
numerical inversion of Laplace transform (NILT) (lossy lines)
Galerkinprojection�
original, stochastic
transmission line
; � < � => > ?> �> ��
��
augmented, deterministic
transmission line
��
� ℜ% AB%CDEFG�
Bfrequency
domain
time domainHB = I + KLB
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Application #1: two wires
Two wires with random position, geometry, and material properties [2]
MNMN
low-level (physical):4 coordinates (x-y) [constrained to avoid overlap]4 geometrical (wire and dielectric radii)
2 material (dielectric permittivity)TOTAL = 10
mid-level (per-unit-length):� = ��� = ��TOTAL = 2
�
�
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Application #1: results
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Application #2: three wires above ground
Three wires with completely random position (sequential placement) [2]
nonparametricproblem!
low-level (physical):N/A
mid-level (per-unit-length):6 per-unit-length inductance entries
6 per-unit-length capacitance entriesTOTAL = 12
�
�
classical PCE N/A !
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Application #2: results
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Conclusions
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Novel hierarchical approach for stochastic analysis of transmission lines
PCE-based modeling w.r.t. mid-level (per-unit-length) parameters
Empirical distribution of per-unit-length parameters fitted using a MoG
Suitable basis functions computed via Gram-Schmidt orthogonalization
PCE coefficients of line response obtained with Galerkin-based simulation
Higher accuracy for a given expansion order
Possible dimensionality reduction (⇒ higher efficiency)
Handling of nonparametric problems (e.g., sequential wire placement)
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References
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[1] P. Manfredi, D. Vande Ginste, I. S. Stievano, D. De Zutter, and F.G.
Canavero, “Stochastic transmission line analysis via polynomial chaos
methods: an overview,” IEEE Electromagn. Compat. Mag., vol. 6, no. 3, pp.
77–84, 2017.
[2] P. Manfredi, “A hierarchical approach to dimensionality reduction and
nonparametric problems in the polynomial chaos simulation of transmission
lines,” IEEE Trans. Electromagn. Compat. (early access).
[3] C. Cui and Z. Zhang, “Stochastic collocation with non-Gaussian
correlated process variations: theory, algorithms and applications,” IEEE
Trans. Compon. Packag. Manuf. Technol. (early access).
Thank you for your attention!
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Application #1: MoG fit & polynomial basis
empirical distribution
MoG fit
number of MoG
components is
increased until convergence
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Application #3: lossy stripline
Stripline interconnect with frequency-dependent conductor losses
PEC
PEC
=PQ = ;�� 00 ;��
=SG = ;′�� ;′$;′$ ;′��
U H, � = =PQ � + H/1� =SG � + H PQ(�)
PQ = ��� �$�$ ���
�PQ = WXW�Y� PQ&�
PEC ground��
reciprocity
MNZ
����
��
��
��
��
��
��
Homgeneous structure ⇒ C-matrix has explicit dependence on PQ!
SPICE model for frequency-dependent per-unit-length impedance matrix
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Application #3: results (5% variation)
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Application #3: results (10% variation)
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Application #3: accuracy & efficiency
error definition: