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# CSE245: Computer-Aided Circuit Simulation and Verification - PowerPoint PPT Presentation

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

Lecture Note 2: State Equations

Spring 2010

Prof. Chung-Kuan Cheng

• 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

• KCL

• KVL

n-1 independent cutsets

m-(n-1) independent loops

• 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

Inductors

v=L(i)di/dt

Mutual inductance

V12=M12,34di34/dt

• 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

• Or we can re-write the equations as:

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

• 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

Output Equation

(suppose v3 is desired output)

State Equation

• State Equation

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

• Where

• 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.

• Definition:

• Simple Transform Pairs

• Laplace Transform Property - Derivatives

• Time Domain State Equation

• Laplace Transform to Frequency Domain

• Re-write the first equation

• Solve for X, we have the frequency domain solution

• 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

• 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

• Re-write Maclaurin Expansion of the state response function

• Moments are

• A voltage or current can be approximated by

• For the state response function, we have

• (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)