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CSE245: Computer-Aided Circuit Simulation and Verification

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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
- 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

- 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
- 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

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

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 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

- 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

Responses in Time Domain

- State Equation

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

- Where

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

- Definition:

- Simple Transform Pairs

- Laplace Transform Property - Derivatives

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 s0, 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

- 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

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)

- (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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