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CSE245: Computer-Aided Circuit Simulation and Verification. Lecture Note 2: State Equations Spring 2010 Prof. Chung-Kuan Cheng. State Equations. Motivation Formulation Analytical Solution Frequency Domain Analysis Concept of Moments. Motivation. Why Whole Circuit Analysis

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cse245 computer aided circuit simulation and verification

CSE245: Computer-Aided Circuit Simulation and Verification

Lecture Note 2: State Equations

Spring 2010

Prof. Chung-Kuan Cheng

state equations
State Equations
  • Motivation
  • Formulation
  • Analytical Solution
  • Frequency Domain Analysis
  • Concept of Moments
motivation
Motivation
  • Why
    • Whole Circuit Analysis
    • Interconnect Dominance
      • Wires smaller  R increase
      • Separation smaller  C increase
  • What
    • Power Net, Clock, Interconnect Coupling, Parallel Processing
  • Where
    • Matrix Solvers, Integration For Dynamic System
    • RLC Reduction, Transmission Lines, S Parameters
    • Whole Chip Analysis
    • Thermal, Mechanical, Biological Analysis
formulation
Formulation
  • General Equation (a.k.a. state equations)
  • Equation Formulation
    • Conservation Laws
      • KCL (Kirchhoff’s Current Law)
        • n-1 equations, n is number of nodes in the circuit
      • KVL (Kirchhoff’s Voltage Law)
        • m-(n-1) equations, m is number of branches in the circuit.
    • Branch Constitutive Equations
      • m equations
formulation1
Formulation
  • State Equations (Modified Nodal Analysis):
  • Desired variables
  • Capacitors: voltage variables
  • Inductors: current variables
  • Current controlled sources: control currents
  • Controlled voltage sources: currents of controlled voltage sources.
  • Freedom of the choices
  • Tree trunks: voltage variables
  • Branches: current variables
conservation laws
Conservation Laws
  • KCL
  • KVL

n-1 independent cutsets

m-(n-1) independent loops

branch constitutive laws
Branch Constitutive Laws
  • Each branch has a circuit element
    • Resistor
    • Capacitor
      • Forward Euler (FE) Approximation
      • Backward Euler (BE) Approximation
      • Trapezoidal (TR) Approximation
    • Inductor
      • Similar approximation (FE, BE or TR) can be used for inductor.

v=R(i)i

i=dq/dt=C(v)dv/dt

branch constitutive laws1
Branch Constitutive Laws

Inductors

v=L(i)di/dt

Mutual inductance

V12=M12,34di34/dt

formulation cutset and loop analysis
Formulation - Cutset and Loop Analysis
  • find a cutset for each trunk
    • write a KCL for each cutset
  • Select tree trunks and links
  • find a loop for each link
    • write a KVL for each loop

cutset matrix

loop matrix

formulation cutset and loop analysis1
Formulation - Cutset and Loop Analysis
  • Or we can re-write the equations as:
  • In general, the cutset and loop matrices can be written as
formulation state equations
Formulation – State Equations
  • From the cutset and loop matrices, we have
  • Combine above two equations, we have the state equation
  • In general, one should
    • Select capacitive branches as tree trunks
      • no capacitive loops
      • for each node, there is at least one capacitor (every node actually should have a shunt capacitor)
    • Select inductive branches as tree links
      • no inductive cutsets
formulation an example
Formulation – An Example

Output Equation

(suppose v3 is desired output)

State Equation

responses in time domain
Responses in Time Domain
  • State Equation
  • The solution to the above differential equation is the time domain response
  • Where
exponential of a matrix
Exponential of a Matrix
  • Calculation of eA is hard if A is large
  • Properties of eA
  • k! can be approximated by Stirling Approximation
  • That is, higher order terms of eA will approach 0 because k! is much larger than Ak for large k’s.
responses in frequency domain laplace transform
Responses in Frequency Domain: Laplace Transform
  • Definition:
  • Simple Transform Pairs
  • Laplace Transform Property - Derivatives
responses in frequency domain
Responses in Frequency Domain
  • Time Domain State Equation
  • Laplace Transform to Frequency Domain
  • Re-write the first equation
  • Solve for X, we have the frequency domain solution
serial expansion of matrix inversion
Serial Expansion of Matrix Inversion
  • For the case s0, assuming initial condition x0=0, we can express the state response function as
  • For the case s, assuming initial condition x0=0, we can express the state response function as
concept of moments
Concept of Moments
  • The moments are the coefficients of the Taylor’s expansion about s=0, or Maclaurin Expansion
  • Recall the definition of Laplace Transform
  • Re-Write as
  • Moments
concept of moments1
Concept of Moments
  • Re-write Maclaurin Expansion of the state response function
  • Moments are
moments calculation an example1
Moments Calculation: An Example
  • A voltage or current can be approximated by
  • For the state response function, we have
moments calculation an example cont d
Moments Calculation: An Example (Cont’d)
  • (1) Set Vs(0)=1 (suppose voltage source is an impulse function)
  • (2) Short all inductors, open all capacitors, derive Vc(0), IL(0)
  • (3) Use Vc(i), IL(i) as sources, i.e. Ic(i+1)=CVc(i) and VL(i+1)=LIL(i), deriveVc(i+1), IL(i+1)
  • (4) i++, repeat (3)
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