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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


  • 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


  • 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


  • 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.



Branch constitutive laws1
Branch Constitutive Laws



Mutual inductance


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)