1. CSE245: Computer-Aided Circuit Simulation and Verification Spring 2006 Chung-Kuan Cheng

2. Administration CK Cheng, CSE 2130, tel. 534-6184, ckcheng@ucsd.educkcheng@ucsd.edu Lectures: 9:30am ~ 10:50am TTH U413A 2 Office Hours: 11:00am ~ 11:50am TTH CSE2130 Textbooks Electronic Circuit and System Simulation Methods T.L. Pillage, R.A. Rohrer, C. Visweswariah, McGraw-Hill Interconnect Analysis and Synthesis CK Cheng, J. Lillis, S. Lin, N. Chang, John Wiley & Sons TA: Vincent Peng (hepeng@cs.ucsd.edu), Rui Shi (rshi@cs.ucsd.edu)hepeng@cs.ucsd.edu

3. Outlines 1.Formulation (2-3 lectures) 2.Linear System (3-4 lectures) 3.Matrix Solver (3-4 lectures) 4.Integration (3-4 lectures) 5.Non-linear System (2-3 lectures) 6.Transmission Lines, S Parameters (2-3 lectures) 7.Sensitivity 8.Mechanical, Thermal, Bio Analysis

4. Grading Homeworks and Projects: 60 Project Presentation: 20% Final Report: 20%

5. Motivation Why –Whole Circuit Analysis, Interconnect Dominance What –Power, Clock, Interconnect Coupling Where –Matrix Solvers, Integration Methods –RLC Reduction, Transmission Lines, S Parameters –Parallel Processing –Thermal, Mechanical, Biological Analysis

6. Circuit Simulation Simulator: Solve CdX/dt=f(X) numerically Input and setup Circuit Output Types of analysis: –DC Analysis –DC Transfer curves –Transient Analysis –AC Analysis, Noise, Distortions, Sensitivity CdX(t)/dt=GX(t)+BU(t) Y=DX(t)+FU(t)

7. Program Structure (a closer look) Numerical Techniques: –Formulation of circuit equations –Solution of ordinary differential equations –Solution of nonlinear equations –Solution of linear equations Input and setup Models Output

8. CSE245: Course Outline Formulation –RLC Linear, Nonlinear Components,Transistors, Diodes –Incident Matrix –Nodal Analysis, Modified Nodal Analysis –K Matrix Linear System –S domain analysis, Impulse Response –Taylor’s expansion –Moments, Passivity, Stability, Realizability –Symbolic analysis, Y-Delta, BDD analysis Matrix Solver –LU, KLU, reordering –Mutigrid, PCG, GMRES

9. CSE245: Course Outline (Cont’) Integration –Forward Euler, Backward Euler, Trapezoidal Rule –Explicit and Implicit Method, Prediction and Correction –Equivalent Circuit –Errors: Local error, Local Truncation Error, Global Error –A-Stable –Alternating Direction Implicit Method Nonlinear System –Newton Raphson, Line Search Transmission Line, S-Parameter –FDTD: equivalent circuit, convolution –Frequency dependent components Sensitivity Mechanical, Thermal, Bio Analysis

10. Lecture 1: Formulation KCL/KVL Sparse Tableau Analysis Nodal Analysis, Modified Nodal Analysis *some slides borrowed from Berkeley EE219 Course

11. Formulation of Circuit Equations Unknowns –B branch currents(i) –N node voltages(e) –B branch voltages(v) Equations –N+B Conservation Laws –B Constitutive Equations

12. Branch Constitutive Equations (BCE) Ideal elements ElementBranch Eqn Resistorv = R·i Capacitori = C·dv/dt Inductorv = L·di/dt Voltage Sourcev = v s, i = ? Current Sourcei = i s, v = ? VCVSv s = A V · v c, i = ? VCCSi s = G T · v c, v = ? CCVSv s = R T · i c, i = ? CCCSi s = A I · i c, v = ?

13. Conservation Laws Determined by the topology of the circuit Kirchhoff’s Voltage Law (KVL): Every circuit node has a unique voltage with respect to the reference node. The voltage across a branch e b is equal to the difference between the positive and negative referenced voltages of the nodes on which it is incident –No voltage source loop Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents flowing out of (or into) any circuit node is zero. –No Current Source Cut

14. Equation Formulation - KCL 0 1 2 R1R1 G2v3G2v3 R3R3 R4R4 I s5 A i = 0 Kirchhoff’s Current Law (KCL) N equations

15. Equation Formulation - KVL 0 1 2 R1R1 G2v3G2v3 R3R3 R4R4 I s5 v - A T e = 0 Kirchhoff’s Voltage Law (KVL) B equations

16. Equation Formulation - BCE 0 1 2 R1R1 G2v3G2v3 R3R3 R4R4 I s5 K v v + i = i s B equations

17. Equation Formulation Node-Branch Incidence Matrix 1 2 3 j B 12iN12iN branches nodesnodes (+1, -1, 0) { A ij = +1 if node i is terminal + of branch j -1 if node i is terminal - of branch j 0 if node i is not connected to branch j

18. Equation Assembly (Stamping Procedures) Different ways of combining Conservation Laws and Constitutive Equations –Sparse Table Analysis (STA) –Modified Nodal Analysis (MNA)

19. Sparse Tableau Analysis (STA) 1.Write KCL: Ai=0 (N eqns) 2.Write KVL:v -A T e=0 (B eqns) 3.Write BCE:K i i + K v v=S(B eqns) N+2B eqns N+2B unknowns N = # nodes B = # branches Sparse Tableau

20. Sparse Tableau Analysis (STA) Advantages It can be applied to any circuit Eqns can be assembled directly from input data Coefficient Matrix is very sparse Problem Sophisticated programming techniques and data structures are required for time and memory efficiency

21. Nodal Analysis (NA) 1.Write KCL A·i=0 (N eqns, B unknowns) 2.Use BCE to relate branch currents to branch voltages i=f(v)(B unknowns  B unknowns) 3.Use KVL to relate branch voltages to node voltages 4.v=h(e)(B unknowns  N unknowns) Y n e=i ns N eqns N unknowns N = # nodes Nodal Matrix

22. Nodal Analysis - Example R3R3 0 1 2 R1R1 G2v3G2v3 R4R4 I s5 1.KCL:Ai=0 2.BCE:K v v + i = i s  i = i s - K v v  A K v v = A i s 3.KVL:v = A T e  A K v A T e = A i s Y n e = i ns

23. Nodal Analysis Example shows NA may be derived from STA Better: Y n may be obtained by direct inspection (stamping procedure) –Each element has an associated stamp –Y n is the composition of all the elements’ stamps

24. Spice input format: Rk N+ N- Rkvalue Nodal Analysis – Resistor “Stamp” N+ N- N+ N- N+ N- i RkRk KCL at node N+ KCL at node N- What if a resistor is connected to ground? …. Only contributes to the diagonal

25. Spice input format: Gk N+ N- NC+ NC- Gkvalue Nodal Analysis – VCCS “Stamp” NC+ NC- N+ N- N+ N- GkvcGkvc NC+ NC- +vc-+vc- KCL at node N+ KCL at node N-

26. Spice input format: Ik N+ N- Ikvalue Nodal Analysis – Current source “Stamp” N+ N- N+ N- N+ N- IkIk

27. Nodal Analysis (NA) Advantages Yn is often diagonally dominant and symmetric Eqns can be assembled directly from input data Yn has non-zero diagonal entries Yn is sparse (not as sparse as STA) and smaller than STA: NxN compared to (N+2B)x(N+2B) Limitations Conserved quantity must be a function of node variable –Cannot handle floating voltage sources, VCVS, CCCS, CCVS

28. Modified Nodal Analysis (MNA) i kl cannot be explicitly expressed in terms of node voltages  it has to be added as unknown (new column) e k and e l are not independent variables anymore  a constraint has to be added (new row) How do we deal with independent voltage sources? i kl k l +- E kl klkl

29. MNA – Voltage Source “Stamp” N+ ikik N- +- EkEk Spice input format: Vk N+ N- Ekvalue 001 00 1 0 N+ N- Branch k N+ N- i k RHS

30. Modified Nodal Analysis (MNA) How do we deal with independent voltage sources? Augmented nodal matrix Some branch currents In general:

31. MNA – General rules A branch current is always introduced as and additional variable for a voltage source or an inductor For current sources, resistors, conductors and capacitors, the branch current is introduced only if: –Any circuit element depends on that branch current –That branch current is requested as output

32. MNA – CCCS and CCVS “Stamp”

33. MNA – An example Step 1: Write KCL i1 + i2 + i3 = 0(1) -i3 + i4 - i5 - i6 = 0(2) i6 + i8 = 0(3) i7 – i8 = 0(4) 0 1 2 G2v3G2v3 R4R4 I s5 R1R1 E S6 -+ R8R8 3 E7v3E7v3 -+ 4 + v 3 - R3R3

34. MNA – An example Step 2: Use branch equations to eliminate as many branch currents as possible 1/R1·v1 + G2 ·v3 + 1/R3·v3 = 0(1) - 1/R3·v3 + 1/R4·v4 - i6 = is5(2) i6 + 1/R8·v8 = 0(3) i7 – 1/R8·v8 = 0(4) Step 3: Write down unused branch equations v6 = ES6(b6) v7 – E7·v3 = 0(b7)

35. MNA – An example Step 4: Use KVL to eliminate branch voltages from previous equations 1/R1·e1 + G2·(e1-e2) + 1/R3·(e1-e2) = 0(1) - 1/R3·(e1-e2) + 1/R4·e2 - i6 = is5(2) i6 + 1/R8·(e3-e4) = 0(3) i7 – 1/R8·(e3-e4) = 0(4) (e3-e2) = ES6(b6) e4 – E7·(e1-e2) = 0(b7)

36. MNA – An example

37. Modified Nodal Analysis (MNA) Advantages MNA can be applied to any circuit Eqns can be assembled directly from input data MNA matrix is close to Y n Limitations Sometimes we have zeros on the main diagonal