CSE245: Computer-Aided Circuit Simulation and Verification

1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture 1: Introduction and Formulation Spring 2008 Chung-Kuan Cheng

2. Administration • CK Cheng, CSE 2130, tel. 534-6184, ckcheng@ucsd.edu • Lectures: 12:30 ~ 1:50pm TTH WLH2205 • 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 • Grading • Homework and Projects: 60% • Project Presentation: 20% • Final Report: 20%

3. CSE245: Course Outline • Formulation (2-3 lectures) • RLC Linear, Nonlinear Components,Transistors, Diodes • Incident Matrix • Nodal Analysis, Modified Nodal Analysis • K Matrix • Linear System (3-4 lectures) • S domain analysis, Impulse Response • Taylor’s expansion • Moments, Passivity, Stability, Realizability • Symbolic analysis, Y-Delta, BDD analysis • Matrix Solver (3-4 lectures) • LU, KLU, reordering • Mutigrid, PCG, GMRES

4. CSE245: Course Outline (Cont’) • Integration (3-4 lectures) • 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 (2-3 lectures) • Newton Raphson, Line Search • Transmission Line, S-Parameter (2-3 lectures) • FDTD: equivalent circuit, convolution • Frequency dependent components • Sensitivity • Mechanical, Thermal, Bio Analysis

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 Input and setup Simulator: Solve numerically Output Circuit Simulation • Types of analysis: • DC Analysis • DC Transfer curves • Transient Analysis • AC Analysis, Noise, Distortions, Sensitivity

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

8. Lecture 1: Formulation • Derive from KCL/KVL • Sparse Tableau Analysis (IBM) • Nodal Analysis, Modified Nodal Analysis (SPICE) *some slides borrowed from Berkeley EE219 Course

9. Conservation Laws • Determined by the topology of the circuit • 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 • Kirchhoff’s Voltage Law (KVL): Every circuit node has a unique voltage with respect to the reference node. The voltage across a branch vb is equal to the difference between the positive and negative referenced voltages of the nodes on which it is incident • No voltage source loop

10. Branch Constitutive Equations (BCE) Ideal elements

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 • 2B+N equations, 2B+N unknowns => unique solution

12. R3 1 2 Is5 R1 R4 G2v3 0 Equation Formulation - KCL Law: State Equation: A i = 0 Node 1: Node 2: N equations Branches Kirchhoff’s Current Law (KCL)

13. Equation Formulation - KVL R3 1 2 Is5 R1 R4 G2v3 0 Law: State Equation: v - AT e = 0 vi = voltage across branch i ei = voltage at node i B equations Kirchhoff’s Voltage Law (KVL)

14. R3 1 2 Is5 R1 R4 G2v3 0 Equation Formulation - BCE Law: State Equation: Kvv + Kii = is B equations

15. 1 2 3 j B 1 2 i N { +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 Aij = Equation FormulationNode-Branch Incidence Matrix A branches n o d e s (+1, -1, 0)

16. Equation Assembly (Stamping Procedures) • Different ways of combining Conservation Laws and Branch Constitutive Equations • Sparse Table Analysis (STA) • Nodal Analysis (NA) • Modified Nodal Analysis (MNA)

17. Sparse Tableau Analysis (STA) • Write KCL: Ai=0 (N eqns) • Write KVL: v - ATe=0 (B eqns) • Write BCE: Kii + Kvv=S (B eqns) N+2B eqns N+2B unknowns N = # nodes B = # branches Sparse Tableau

18. 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 Disadvantages • Sophisticated programming techniques and data structures are required for time and memory efficiency

19. Nodal Analysis (NA) 1. Write KCL Ai=0 (N equations, B unknowns) 2. Use BCE to relate branch currents to branch voltages i=f(v) (B equations  B unknowns) • Use KVL to relate branch voltages to node voltages v=h(e) (B equations  N unknowns) N eqns N unknowns N = # nodes Yne=ins Nodal Matrix

20. 1 2 Is5 R1 R4 G2v3 0 Nodal Analysis - Example R3 • KCL: Ai=0 • BCE: Kvv + i = is i = is - Kvv  A Kvv = A is • KVL: v = ATe  A KvATe = A is Yne = ins Yn = AKvAT Ins = Ais

21. Nodal Analysis • Example shows how NA may be derived from STA • Better Method: Yn may be obtained by direct inspection (stamping procedure) • Each element has an associated stamp • Yn is the composition of all the elements’ stamps

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

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

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

25. 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

26. Modified Nodal Analysis (MNA) How do we deal with independent voltage sources? • ikl cannot be explicitly expressed in terms of node voltages  it has to be added as unknown (new column) • ek and el are not independent variables anymore  a constraint has to be added (new row) Ekl k l + - l k ikl

27. N+ N- ik RHS Ek + - N+ N- Branch k N- ik MNA – Voltage Source “Stamp” Spice input format: Vk N+ N- Ekvalue N+

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

29. MNA – General rules • A branch current is always introduced as an 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

30. MNA – CCCS and CCVS “Stamp”

31. + v3 - ES6 R3 - + 3 1 2 R8 Is5 R1 R4 G2v3 - + 0 4 E7v3 MNA – An example Step 1: Write KCL (1) (2) (3) (4)

32. MNA – An example Step 2: Use branch equations to eliminate as many branch currents as possible (1) (2) (3) (4) Step 3: Write down unused branch equations (b6) (b7)

33. MNA – An example Step 4: Use KVL to eliminate branch voltages from previous equations (1) (2) (3) (4) (b6) (b7)

34. MNA – An example

35. 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 Yn Limitations • Sometimes we have zeros on the main diagonal