CSE245: Computer-Aided Circuit Simulation and Verification

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: n-1 tree trunks and m-n+1 links, n: #nodes, m: #edges. • Conservation Laws • KCL (Kirchhoff’s Current Law: n-1 cuts on tree trunk) • n-1 equations, n is number of nodes in the circuit • KVL (Kirchhoff’s Voltage Law: m-n+1 loops on the links) • m-(n-1) equations, • Branch Constitutive Equations • m equations

6. Formulation State Equations (Modified Nodal Analysis): Desired variables (branch elements) • Capacitors: voltage variables • Inductors: current variables • Current/voltage controlled sources: control current/voltage • Controlled sources: currents of controlled voltage sources, and voltages of controlled current sources Freedom of the choices (topology) • Tree trunks: voltage variables • 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 More examples to demonstrate E, -ET relation

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)

26. 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) Use an example to illustrate the Elmore delay model

27. Serial Expansion of Matrix Inversion • For the case s0, assuming initial condition x0=0, we can express the state response function as Exercise: Show that x=-A-1Bu-A-1CA-1Bdu/dt-(A-1C)2A-1Bd2u /dt2-… is a solution of the ODE problem, Cdx/dt=Ax+Bu. Assumption: A-1 exists Dku/dtk shrinks to 0 as k increases