Lecture Notes on Computer-Aided Circuit Simulation and Verification: State Equations, Slides of Computer Science

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CSE245: Computer-Aided Circuit

Simulation and Verification

Lecture Note 2: State Equations

State Equations

1. Motivation

2. Formulation

3. Analytical Solution

4. Frequency Domain

Analysis

5. Concept of Moments

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

Formulation

State Equations (Modified Nodal Analysis):

Desired variables

1.Capacitors: voltage variables

2.Inductors: current variables

3.Current controlled sources: control currents

4.Controlled voltage sources: currents of controlled

voltage sources.

Freedom of the choices

1.Tree trunks: voltage variables

2.Branches: current variables

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 Laws

Inductors

v=L(i)di/dt

Mutual inductance

V 12 =M12,34 di 34 /dt

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

• From the cutset and loop matrices, we have
  • 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
• Combine above two equations, we have the state equation

Responses in Time Domain

• State Equation
  • The solution to the above differential equation is the time domain response
• Where

Exponential of a Matrix

• Properties of e A
  • k! can be approximated by Stirling Approximation
• That is, higher order terms of e A^ will approach 0 because k! is much larger than A k^ for large k’s.
• Calculation of e A^ is hard if A is large

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

• For the case s0, assuming initial condition x 0 =0, we can express the state response function as
• For the case s∞, assuming initial condition x 0 =0, we can express the state response function as

Concept of Moments

• Re-write Maclaurin Expansion of the state response function
• Moments are

Moments Calculation: An Example

• For the state response function, we have
• A voltage or current can be approximated by