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

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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
Conservation Laws
• KCL
• KVL

n-1 independent cutsets

m-(n-1) independent loops

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

V12=M12,34di34/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
Formulation – An Example

Output Equation

(suppose v3 is desired output)

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