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CSE245: Computer-Aided Circuit Simulation and Verification

CSE245: Computer-Aided Circuit Simulation and Verification. Lecture Note 2: State Equations Prof. Chung-Kuan Cheng. State Equations. Motivation Formulation Analytical Solution Frequency Domain Analysis Concept of Moments. Motivation. Why Whole Circuit Analysis Interconnect Dominance

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CSE245: Computer-Aided Circuit Simulation and Verification

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  1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture Note 2: State Equations Prof. Chung-Kuan Cheng

  2. State Equations • Motivation • Formulation • Analytical Solution • Frequency Domain Analysis • Concept of Moments

  3. 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

  4. Formulation • Nodal Analysis • Link Analysis • Modified Nodal Analysis • Regularization

  5. 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

  6. 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

  7. 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

  8. Nodal Analysis

  9. 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.

  10. 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

  11. Formulation - Cutset and Loop Analysis • Or we can re-write the equations as: • In general, the cutset and loop matrices can be written as

  12. 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

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

  14. 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

  15. Branch Constitutive Laws Inductors v=L(i)di/dt Mutual inductance V12=M12,34di34/dt

  16. Responses in Time Domain • State Equation • The solution to the above differential equation is the time domain response • Where

  17. 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.

  18. Responses in Frequency Domain: Laplace Transform • Definition: • Simple Transform Pairs • Laplace Transform Property - Derivatives

  19. 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

  20. 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

  21. 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

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

  23. Moments Calculation: An Example

  24. Moments Calculation: An Example • A voltage or current can be approximated by • For the state response function, we have

  25. 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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