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CSE245: Computer-Aided Circuit Simulation and Verification

Dive into state equations formulation for circuit analysis, including conservation laws, nodal analysis, and branch constitutive laws. Explore cutset and loop analysis for detailed circuit verification.

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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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