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CSE245: Computer-Aided Circuit Simulation and Verification. Lecture Notes 3 Model Order Reduction (1) Spring 2010 Prof. Chung-Kuan Cheng. Outline. Introduction Formulation Linear System Time Domain Analysis Frequency Domain Analysis Moments Stability and Passivity

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cse245 computer aided circuit simulation and verification

CSE245: Computer-Aided Circuit Simulation and Verification

Lecture Notes 3

Model Order Reduction (1)

Spring 2010

Prof. Chung-Kuan Cheng

outline
Outline
  • Introduction
  • Formulation
  • Linear System
    • Time Domain Analysis
    • Frequency Domain Analysis
    • Moments
    • Stability and Passivity
    • Model Order Reduction
transfer function
Transfer Function

State Equation in Frequency Domain ( suppose zero initial condition)

  • Recall:
  • U is 1xn vector of inputs (voltage, current sources)
  • X is state variable (voltage or current of inductor, cap., etc.)
  • Y is voltages of current sources & currents of voltage sources.
  • C is defined as
  • A is defined as
  • Will show that D = BT

Solve X

Express Y(s) as

a function of U(s)

Transfer Function:

stability
Stability
  • A network is stable if, for all bounded inputs, the output is bounded.
  • For a stable network, transfer function H(s)=N(s)/D(s) (general form)

(Recall: H(s) = Y(s)/U(s) in our case)

    • Should have only negative poles pj, i.e. Re(pj)  0
    • If pole falls on the imaginary axis, i.e. Re(pi) = 0, it must be a simple pole.
passivity
Passivity
  • Passivity
    • Passive system doesn’t generate energy
    • A one-port network is said to be passive if the total power dissipation is nonnegative for all initial time t0, for all time t>t0, and for all possible input waveforms, that is,

where E(t0) is the energy stored at time t0

  • Passivity of a multi-port network
    • If all elements of the network are passive, the network is passive
passivity in complex space representation
Passivity in Complex Space Representation
  • For steady state response of a one-port
  • The complex power delivered to this one-port is
  • For a passive network, the average power delivered to the network should be nonnegative
linear multi port passivity
Linear Multi-Port Passivity
  • For multi-port, suppose each port is either a voltage source or a current source
    • For a voltage source port, the input is the voltage and the output is a current
    • For a current source port, the input is the current and the output is a voltage
    • Then we will have D=BT in the state equation
    • Let U(s) be the input vector of all ports, and H(s) be the transfer function, thus the output vector Y(s) = H(s)U(s)
  • Average power delivered to this multi-port network is
  • For a passive network, we should have
linear system passivity
Linear System Passivity
  • State Equation (s domain)
  • We have shown that transfer function is

and

where

  • We will show that this network is passive, that is

…Why is this important?

After Model Order Reduction (MOR) (in later slides), this condition must still be true.

Plug conjugate H(s) in here

passivity proof part 1
Passivity Proof (part 1)
  • To show
  • Is equivalent to show

Here’s how…

where

Re[ U*(s)H(s)*U(s) ] ≥ 0

Set U*(s)H(s)*U(s) = F

½ Re(F* + F) ≥ 0

F* + F ≥ 0

Plugging F back in:

(U*(s)H*(s)U(s))* + U*(s)H*(s)U(s)

= U*(s)H(s)U(s) + U*(s)H*(s)U(s)

= U*(s) (H(s) + H*(s)) U(s) ≥ 0

passivity proof part 2
Passivity Proof (part 2)

_

The s and s combine

The terms E and –ET cancel

  • We have
  • Thus
passivity and stability
Passivity and Stability
  • A passive network is stable.
  • However, a stable network is not necessarily passive.
      • All poles could be on LHS, but some could be negative!
  • A interconnect network of stable components is not necessarily stable.
  • The interconnection of passive components is passive.
model order reduction mor
Model Order Reduction (MOR)
  • MOR techniques are used to build a reduced order model to approximate the original circuit

Huge

Network

Formulation

Small

Network

Realization

MOR

model order reduction overview
Model Order Reduction: Overview
  • Explicit Moment Matching
    • AWE, Pade Approximation
  • Implicit Moment Matching
    • Krylov Subspace Methods
      • PRIMA, SPRIM
  • Gaussian Elimination
    • TICER
    • Y-Delta Transformation
moments review
Moments Review
  • Transfer function
  • Compare
  • Moments
moments matching pade approximation
Moments Matching: Pade Approximation

Choose the 2q rational function coefficients

So that the reduced rational function matches the first 2q moments

of the original transfer function H(s).

moments matching pade approximation1
Moments Matching: Pade Approximation
  • Step 1: calculate the first 2q moments of H(s)
  • Step 2: calculate the 2q coeff. of the Pade’ approximation, matching the first 2q moments of H(s)
pade approximation coefficients
Pade Approximation: Coefficients

For a1 a2,…, aq solve the following linear system:

Then, use the a1 a2 … aq to calculate b0 b1 … bq-1 :

pade approximation drawbacks
Pade Approximation: Drawbacks
  • Numerically unstable
    • Higher order moments
  • Matrix powers converge to the eigenvector corresponding to the largest eigenvalue.
  • Columns become linear dependent for large q. The problem is numerically very ill-conditioned.
  • Passivity is not always preserved.
    • Pade may generate positive poles