CSE245: Computer-Aided Circuit Simulation and Verification

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CSE245 Computer-Aided Circuit Simulation and
Verification

• Lecture Note 2 State Equations
• Prof. Chung-Kuan Cheng

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State Equations

1. Motivation
2. Formulation
3. Analytical Solution
4. Frequency Domain Analysis
5. Concept of Moments

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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

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Formulation

• Nodal Analysis
• Link Analysis
• Modified Nodal Analysis
• Regularization

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Formulation

• General Equation (a.k.a. state equations)
• Equation Formulation
• Conservation Laws
• KCL (Kirchhoffs Current Law)
• n-1 equations, n is number of nodes in the
  circuit
• KVL (Kirchhoffs Voltage Law)
• m-(n-1) equations, m is number of branches in the
  circuit.
• Branch Constitutive Equations
• m equations

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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
• Tree links current variables

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Conservation Laws

• KCL Cut is related to each trunk and links
• KVL Loop is related to each link and the trunks

n-1 independent cutsets
m-(n-1) independent loops
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Nodal Analysis

•  

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Link Analysis

• Variables link currents
• Equations KVL of loops formed by each link and
  tree trunks.
• Example Provide an example of the formula
• Remark The system matrix is symmetric and
  positive definite.

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Formulation - Cutset and Loop Analysis

• Select tree trunks and links

• find a cutset for each trunk
• write a KCL for each cutset

• find a loop for each link
• write a KVL for each loop

cutset matrix
loop matrix
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Formulation - Cutset and Loop Analysis

• Or we can re-write the equations as

• In general, the cutset and loop matrices can be
  written as

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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

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Formulation An Example
Output Equation (suppose v3 is desired output)
State Equation
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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.

vR(i)i
idq/dtC(v)dv/dt
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Branch Constitutive Laws
Inductors
vL(i)di/dt
Mutual inductance
V12M12,34di34/dt
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Responses in Time Domain

• State Equation

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

• Where

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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 ks.

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Responses in Frequency Domain Laplace Transform

• Definition

• Simple Transform Pairs

• Laplace Transform Property - Derivatives

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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

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Serial Expansion of Matrix Inversion

• For the case s?0, assuming initial condition
  x00, we can express the state response function
  as

• For the case s??, assuming initial condition
  x00, we can express the state response function
  as

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Concept of Moments

• The moments are the coefficients of the Taylors
  expansion about s0, or Maclaurin Expansion

• Recall the definition of Laplace Transform

• Re-Write as

• Moments

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Concept of Moments

• Re-write Maclaurin Expansion of the state
  response function

• Moments are

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Moments Calculation An Example
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Moments Calculation An Example

• A voltage or current can be approximated by

• For the state response function, we have

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Moments Calculation An Example (Contd)

• (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(i1)CVc(i) and VL(i1)LIL(i), derive
  Vc(i1), IL(i1)
• (4) i, repeat (3)