1. CSE 245: Computer Aided Circuit Simulation and Verification Instructor: Prof. Chung-Kuan Cheng Winter 2003 Lecture 1: Formulation

2. Jan. 24, 2003Cheng & Zhu, UCSD @ 20032 Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

3. Jan. 24, 2003Cheng & Zhu, UCSD @ 20033 History of SPICE  SPICE -- Simulation Program with Integrated Circuit Emphasis  1969, CANCER developed by Laurence Nagel on Prof. Ron Roher ’ s class  1970~1972, CANCER program  May 1972, SPICE-I release  July ’ 75, SPICE 2A, …, 2G  Aug 1982, SPICE 3 (in C language)  No new progress on software package since then

4. Jan. 24, 2003Cheng & Zhu, UCSD @ 20034 RCL circuit

5. Jan. 24, 2003Cheng & Zhu, UCSD @ 20035 RCL circuit (II)  General Circuit Equation  Consider homogeneous form first Q: How to Compute A k ? and

6. Jan. 24, 2003Cheng & Zhu, UCSD @ 20036  Assume A has non-degenerate eigenvalues and corresponding linearly independent eigenvectors, then A can be decomposed as where and Solving RCL Equation

7. Jan. 24, 2003Cheng & Zhu, UCSD @ 20037  What ’ s the implication then?  To compute the eigenvalues: Solving RCL Equation (II) real eigenvalue Conjugative Complex eigenvalue where

8. Jan. 24, 2003Cheng & Zhu, UCSD @ 20038 Solving RCL Equation (III) In the previous example where hence Let c=r=l=1, we have

9. Jan. 24, 2003Cheng & Zhu, UCSD @ 20039  What if matrix A has degenerated eigenvalues? Jordan decomposition ! Solving RCL Equation (IV) J is in the Jordan Canonical form And still

10. Jan. 24, 2003Cheng & Zhu, UCSD @ 200310 Jordan Decomposition similarly

11. Jan. 24, 2003Cheng & Zhu, UCSD @ 200311 Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

12. Jan. 24, 2003Cheng & Zhu, UCSD @ 200312 Equation Formulation  KCL Converge of node current  KVL Closure of loop voltage  Brach equations I, R relations

13. Jan. 24, 2003Cheng & Zhu, UCSD @ 200313 Types of elements  Resistor  Capacitor  Inductor L is even dependent on frequency due to skin effect, etc …  Controlled Sources VCVS, VCCS, CCVS, CCCS

14. Jan. 24, 2003Cheng & Zhu, UCSD @ 200314 Cut-set analysis 1. Construct a spanning tree 2. Take as much capacitor branches as tree branches as possible 3. Derive the fundamental cut-set, in which each cut truncates exactly one tree branch 4. Write KCL equations for each cut 5. Write KVL equations for each tree link 6. Write the constitution equation for each branch

15. Jan. 24, 2003Cheng & Zhu, UCSD @ 200315 KCL Formulation #nodes-1 lines #braches columns

16. Jan. 24, 2003Cheng & Zhu, UCSD @ 200316 KCL Formulation (II)  Permute the columns to achieve a systematic form

17. Jan. 24, 2003Cheng & Zhu, UCSD @ 200317 KVL Formulation Remove the equations for tree braches and systemize

18. Jan. 24, 2003Cheng & Zhu, UCSD @ 200318 Cut & Loop relation In the previous example

19. Jan. 24, 2003Cheng & Zhu, UCSD @ 200319 Sparse Tableau Analysis (STA)  n=#nodes, b=#branches (n-1) KCL b KVL b branch relations b b n-1 Totally 2b+n-1 variables, 2b+n-1 equations bbn-1 Due to independent sources

20. Jan. 24, 2003Cheng & Zhu, UCSD @ 200320 STA (II)  Advantages Covers any circuit Easy to assemble Very sparse  K i, K v, I each has exactly b non-zeros. A and A T each has at most 2b non-zeros.  Disadvantages Sophisticated data structures & programming techniques

21. Jan. 24, 2003Cheng & Zhu, UCSD @ 200321 Agenda  RCL Network  Sparse Tableau Analysis  Modified Nodal Analysis

22. Jan. 24, 2003Cheng & Zhu, UCSD @ 200322 Nodal Analysis  Derivation From STA: (1) (2) (3) (3) x K i -1  (4) x A  (4) Using (a)  (5) (6) Tree trunk voltages Substitute with node voltages (to a given reference), we get the nodal analysis equations.

23. Jan. 24, 2003Cheng & Zhu, UCSD @ 200323 Nodal Analysis (II)

24. Jan. 24, 2003Cheng & Zhu, UCSD @ 200324 Modified Nodal Analysis  General Form Node Conductance matrix KCL Independent current source Independent voltage source Due to non-conductive elements  Y n can be easily derived  Add extra rows/columns for each non-conductive elements using templates

25. Jan. 24, 2003Cheng & Zhu, UCSD @ 200325 MNA (II)  Fill Y n matrix according to incidence matrix Choose n 6 as reference node

26. Jan. 24, 2003Cheng & Zhu, UCSD @ 200326 MNA Templates Independent current source Independent voltage source Add to the right-hand side of the equation

27. Jan. 24, 2003Cheng & Zhu, UCSD @ 200327 MNA Templates (II) CCVS CCCS

28. Jan. 24, 2003Cheng & Zhu, UCSD @ 200328 MNA Templates (III) VCVS VCCS + - + -

29. Jan. 24, 2003Cheng & Zhu, UCSD @ 200329 MNA Templates (IV) M Mutual inductance Operational Amplifier

30. Jan. 24, 2003Cheng & Zhu, UCSD @ 200330 MNA Example Circuit Topology MNA Equations

31. Jan. 24, 2003Cheng & Zhu, UCSD @ 200331 MNA Summary  Advantages Covers any circuits Can be assembled directly from input data. Matrix form is close to Y n  Disadvantages We may have zeros on the main diagonal. Principle minors could be singular