1. Formulation of Circuit Equations Lecture 2 Alessandra Nardi Thanks to Prof. Sangiovanni-Vincentelli and Prof. Newton

2. 219A: Course Overview Fundamentals of Circuit Simulation –Approximately 12 lectures Analog Circuits Simulation –Approximately 4 lectures Digital Systems Verification –Approximately 3 lectures Physical Issues Verification –Approximately 6 lectures E.g.: SPICE, HSPICE, PSPICE, SPECTRE, ELDO ….

3. SPICE history Prof. Pederson with “a cast of thousands” 1969-70: Prof. Roher and a class project –CANCER: Computer Analysis of Nonlinear Circuits, Excluding Radiation 1970-72: Prof. Roher and Nagel –Develop CANCER into a truly public-domain, general-purpose circuit simulator 1972: SPICE I released as public domain –SPICE: Simulation Program with Integrated Circuit Emphasis 1975: Cohen following Nagel research –SPICE 2A released as public domain 1976 SPICE 2D New MOS Models 1979 SPICE 2E Device Levels (R. Newton appears) 1980 SPICE 2G Pivoting (ASV appears)

4. Circuit Simulation Simulator: Solve dx/dt=f(x) numerically Input and setup Circuit Output Types of analysis: –DC Analysis –DC Transfer curves –Transient Analysis –AC Analysis, Noise, Distorsion, Sensitivity

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

6. Formulation of Circuit Equations Circuit with B branches N nodes Simulator Set of equations Set of unknowns

7. 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

8. Branch Constitutive Equations (BCE) Determined by the mathematical model of the electrical behavior of a component –Example: V=R·I In most of circuit simulators this mathematical model is expressed in terms of ideal elements

9. Ideal Elements: Reference Direction Branch voltages and currents are measured according to the associated reference directions –Also define a reference node (ground) + _ v i Two-terminal + _ v1v1 i1i1 Two-port i1i1 + _ v2v2 i2i2 i2i2

10. 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 = ?

11. 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 Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents flowing out of (or into) any circuit node is zero.

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

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

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

15. 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 PROPERTIES A is unimodular 2 nonzero entries in each column

16. Equation Assembly (Stamping Procedures) Different ways of combining Conservation Laws and Constitutive Equations –Sparse Table Analysis (STA) Brayton, Gustavson, Hachtel –Modified Nodal Analysis (MNA) McCalla, Nagel, Roher, Ruehli, Ho

17. 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

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 Problem Sophisticated programming techniques and data structures are required for time and memory efficiency

19. 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

20. 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

21. 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

22. 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

23. 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-

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

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) 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 kl +- E kl klkl

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

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

29. 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

30. MNA – CCCS and CCVS “Stamp”

31. 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

32. 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)

33. 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)

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 Y n Limitations Sometimes we have zeros on the main diagonal and principle minors may also be singular.