1. Probability-Weighted Passivity Enforcement Technique for Fitted Rational Transfer Functions
Joshua Yun Keat Choo
Mentor: Xu Chen Advisor: Andreas Cangellaris
Motivation
Methodology
Random Perturbation
Fitting collected data into a Rational Transfer function allows for the usage of recursive convolution in time domain simulation.
𝑦 𝑘 ≈ 𝛼 1 𝑦 𝑘−1 + 𝛽 0 𝑥 𝑘 + 𝛽 1 𝑥 𝑘−1
Advantages
Recursive Convolution is more efficient. Saves on computing time and memory.
Disadvantages
After performing Vector Fitting, the resulting transfer function may not be passive
Passivity implies causality and stability, hence it is an important criteria if we want to simulate an actual system.
S-Parameters
Our proposed method involves perturbing the poles of the fitted pole-residue model by adding Gaussian magnitude and uniform phase to the poles.
When collecting data, there will inevitably be uncertainties in the data collected due to the imperfect sampling of the device used. Hence, we want to model this uncertainty in the form of a random variable, and we will end up with a passive model with less error.
Vector Fitting [1]
Rational Transfer Function
Passive [2]
Not Passive
Ready for time domain simulation
Passivity Enforcement [3]
Passivity
Stability
Causality
Passivity enforcement may introduce additional errors to the fitted transfer function [1], hence we propose a probability-weighted passivity enforcement technique.
Fitted Poles
Perturbed Poles
Passivity
Passivity Check
Results
For a given system with transfer matrix 𝐻(𝑠) and state-space realization 𝐴,𝐵,𝐶,𝐷 , it is passive if:
𝐴 is asymptotically stable
∥𝐷 ∥ 2 <1
Hamiltonian matrix, 𝑀, has no purely imaginary eigenvalues
For a given state-space matrix characterization, the Hamiltonian matrix 𝑀 is given below:
𝐴+𝐵 𝕀− 𝐷 𝑇 𝑇 −1 𝐷 𝑇 𝐶 𝐵 𝕀− 𝐷 𝑇 𝑇 −1 𝐵 𝑇 − 𝐶 𝑇 𝕀−𝐷 𝐷 𝑇 −1 𝐶 − 𝐴 𝑇 − 𝐶 𝑇 𝐷 𝕀− 𝐷 𝑇 𝐷 −1 𝐵 𝑇
The set of frequencies 𝜔 𝑘 corresponding to the imaginary eigenvalues of 𝑀:
𝜒={ 𝜔 𝑘 :𝑗 𝜔 𝑘 ∈𝜆 𝑀 , 𝜔 𝑘 ≥0 }
However, Condition 3 is not a necessary and sufficient condition for passivity. Even if there are purely imaginary eigenvalues in 𝑀, we can further analyze:
Local passivity in each subband ( 𝜔 𝑘 , 𝜔 𝑘+1 ) is assessed by finding the maximum singular value of the transfer matrix at any frequency point in the subband.
𝜉 𝑘 = 𝜔 𝑘 + 𝜔 𝑘+1 2 ,𝑘= 𝑘 0 ,…,𝐾
𝜂 𝑘 =𝜎 𝑗 𝜉 𝑘 = max 𝜎(𝐻(𝑗 𝜉 𝑘 ))
The system is locally passive in ( 𝜔 𝑘 , 𝜔 𝑘+1 ) if and only if 𝜂 𝑘 <1.
The system is locally not passive in ( 𝜔 𝑘 , 𝜔 𝑘+1 ) if and only if 𝜂 𝑘 >1.
Since the physical system will give us a symmetric 𝑆(𝑠) for the S-Parameters, we will end up with a pole-residue model with symmetric residue matrices, 𝑅 𝑚 and a symmetrical 𝐷.
𝑆 𝑠 = 𝑚= 𝑁 𝑅 𝑚 𝑠− 𝑎 𝑚 +𝐷+𝑠𝐸
We are then able to use the “Passivity Matrix”, 𝑃, which via the subset of negative-real eigenvalues − 𝜔 2 , give the frequencies 𝑗𝜔, the boundaries of passivity violations.
𝑃=(𝐴−𝐵 𝐷−𝕀 −1 𝐶)(𝐴−𝐵 𝐷+𝕀 −1 𝐶)
Advantages
The passivity matrix is half the size of the Hamiltonian matrix. Reduction in time needed by factor of nearly eight.
Disadvantages
Restricted to fitting a symmetrical pole-residue model
I wrote a code that uses this method to check for passivity in a system.
Existing passivity enforcement algorithm changes transfer function and introduces a large error. We will continue to investigate how we can use stochastic methods to get a fit that is both passive and realistic. We want to get the model that will most likely be correct, based on a probability measure.
Below are some plots to show that a fitted rational function will have less errors than a passivity enforced fitted rational function.
𝑆 12 Parameters before Passivity Enforcement
𝑆 12 Parameters after Passivity Enforcement
Reference
[1] S. Grivet-Talocia, B. Gustavsen. Passive Macromodeling: Theory and Applications. Hoboken, NJ: John Wiley & Sons, Inc., 2016.
[2] B. Gustavsen and A. Semlyen, “Fast passivity assessment for s-parameter rational models via a half-size test matrix,” in IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 1, pp , Dec
[3] S. Grivet-Talocia, "Enforcing passivity of macromodels via spectral perturbation of hamiltonian matrices,” in Signal Propagation on Interconnects (SPI), th IEEE Workshop on, pp , May 2003.