CSE245: Computer-Aided Circuit Simulation and Verification

Lecture 1: Introduction and Formulation

Spring 2010

Chung-Kuan Cheng

Administration

• CK Cheng, CSE 2130, tel. 534-6184, ckcheng@ucsd.edu

• Lectures: 3:30 ~ 4:50pm TTH CSB005• Textbooks

– Required: Electronic Circuit and System Simulation MethodsT.L. Pillage, R.A. Rohrer, C. Visweswariah, McGraw-Hill

– Recommended: Interconnect Analysis and SynthesisCK Cheng, J. Lillis, S. Lin, N. Chang, John Wiley & Sons

• Grading– Homework and Projects: 60%– Project Presentation: 20%– Final Report: 20%

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

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 (3-4 lectures)– Direct Method, Adjoint Network Method

• Mechanical, Thermal, Bio Analysis

Motivation: Analysis• Energy: Fission, Fusion, Fossil Energy,

Efficiency Optimization

• Astrophysics: Dark energy, Nucleosynthesis

• Climate: Pollution, Weather Prediction

• Biology: Microbial life

• Socioeconomic Modeling: Global scale modeling

Nonlinear Systems, ODE, PDE, Heterogeneous Systems, Multiscale Analysis.

Motivation: Analysis

Exascale Calculation: 1018, Parallel Processing at 20MW

• 10 Petaflops 2012

• 100 PF 2017

• 1000 PF 2022

Motivation: Analysis

• Modeling: Inputs, outputs, system models.

• Simulation: Time domain, frequency domain, wavelet simulation.

• Sensitivity Calculation: Optimization

• Uncertainty Quantification: Derivation with partial information or variations

• User Interface: Data mining, visualization,

Motivation: Circuit Analysis

• 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

Circuit Simulation

Simulator:Solve numerically

Input and setup Circuit

Output

Types of analysis:– DC Analysis– DC Transfer curves– Transient Analysis– AC Analysis, Noise, Distortions, Sensitivity

)()(

)()()(

)(

tFUtDXY

tBUtGXdt

tdXCXf

dt

tdXCXf

)()(

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

Lecture 1: Formulation

• Derive from KCL/KVL

• Sparse Tableau Analysis (IBM)

• Nodal Analysis, Modified Nodal Analysis (SPICE)

*some slides borrowed from Berkeley EE219 Course

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

Branch Constitutive Equations (BCE)

Ideal elements

Element Branch Eqn Variable parameter

Resistor v = R·i -

Capacitor i = C·dv/dt -

Inductor v = L·di/dt -

Voltage Source v = vs i = ?

Current Source i = is v = ?

VCVS vs = AV · vc i = ?

VCCS is = GT · vc v = ?

CCVS vs = RT · ic i = ?

CCCS is = AI · ic v = ?

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

Equation Formulation - KCL

0

1 2

R1G2v3

R3

R4Is5

0

0

11100

00111

5

4

3

2

1

i

i

i

i

iA i = 0

Kirchhoff’s Current Law (KCL)

N equations

Law: State Equation:

Node 1:Node 2:

Branches

Equation Formulation - KVL

0

1 2

R1G2v3

R3

R4Is5

0

0

0

0

0

10

10

11

01

01

2

1

5

4

3

2

1

e

e

v

v

v

v

vv - AT e = 0

Kirchhoff’s Voltage Law (KVL)

B equations

Law: State Equation:

vi = voltage across branch iei = voltage at node i

Equation Formulation - BCE

0

1 2

R1G2v3

R3

R4Is5

55

4

3

2

1

5

4

3

2

1

4

3

2

1

0

0

0

0

00000

01

000

001

00

0000

00001

sii

i

i

i

i

v

v

v

v

v

R

R

GR

Kvv + Kii = is

B equations

Law: State Equation:

Equation FormulationNode-Branch Incidence Matrix A

1 2 3 j B

12

i

N

branches

nodes (+1, -1, 0)

{Aij = +1 if node i is + terminal of branch j-1 if node i is - terminal of branch j0 if node i is not connected to branch j

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)

Sparse Tableau Analysis (STA)

1. Write KCL: Ai=0 (N eqns)

2. Write KVL: v - ATe=0 (B eqns)

3. Write BCE: Kii + Kvv=S (B eqns)

Se

v

i

KK

AI

A

vi

T 0

0

0

0

00 N+2B eqnsN+2B unknowns

N = # nodesB = # branches

Sparse Tableau

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

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)

3. Use KVL to relate branch voltages to node voltages

v=h(e) (B equations N unknowns)

Yne=ins

N eqnsN unknownsN = # nodes

Nodal Matrix

Nodal Analysis - ExampleR3

0

1 2

R1G2v3

R4Is5

1. KCL: Ai=02. BCE: Kvv + i = is i = is - Kvv A Kvv = A is

3. KVL: v = ATe A KvATe = A is

Yne = ins

52

1

433

32

32

10

111

111

sie

e

RRR

RG

RG

RYn = AKvAT

Ins = Ais

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

Spice input format: Rk N+ N- Rkvalue

Nodal Analysis – Resistor “Stamp”

kk

kk

RR

RR11

11N+ N-

N+

N-

N+

N-

iRk

sNNk

others

sNNk

others

ieeR

i

ieeR

i

1

1KCL at node N+

KCL at node N-

What if a resistor is connected to ground?

….Only contributes to the

diagonal

Spice input format: Gk N+ N- NC+ NC- Gkvalue

Nodal Analysis – VCCS “Stamp”

kk

kk

GG

GGNC+ NC-

N+

N-

N+

N-

Gkvc

NC+

NC-

+

vc

-

sNCNCkothers

sNCNCkothers

ieeGi

ieeGi KCL at node N+

KCL at node N-

Spice input format: Ik N+ N- Ikvalue

Nodal Analysis – Current source “Stamp”

k

k

I

IN+ N-

N+

N-

N+

N-

Ik

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

Modified Nodal Analysis (MNA)

• 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)

How do we deal with independent voltage sources?

ikl

k l

+ -Ekl

klkl

l

k

Ei

e

e

011

1

1k

l

MNA – Voltage Source “Stamp”

N+

ik

N-

+ -Ek

Spice input format: Vk N+ N- Ekvalue

kE

0

00 0 1

0 0 -1

1 -1 0

N+

N-

Branch k

N+ N- ik RHS

Modified Nodal Analysis (MNA)

How do we deal with independent voltage sources?

Augmented nodal matrix

MSi

e

C

BYn

0

Some branch currents

MSi

e

DC

BYn

In general:

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

MNA – CCCS and CCVS “Stamp”

MNA – An example

Step 1: Write KCL

0

1 2

G2v3

R4Is5R1

ES6- +

R8

3

E7v3

- +4

+ v3 -R3

8

0

0

0

87

86

6543

321

ii

ii

iiii

iii (1)(2)(3)(4)

MNA – An exampleStep 2: Use branch equations to eliminate as many branch currents as possible

01

01

11

011

88

7

88

6

5644

33

33

3211

vR

i

vR

i

iivR

vR

vR

vGvR

s

Step 3: Write down unused branch equations

(1)

(2)

(3)

(4)

0377

66

vEv

ESv (b6)(b7)

MNA – An exampleStep 4: Use KVL to eliminate branch voltages from previous equations

0)(

)(

0)(1

0)(1

1)(

1

0)(1

)(1

2174

623

438

7

438

6

5624

213

213

21211

eeEe

ESee

eeR

i

eeR

i

iieR

eeR

eeR

eeGeR

s

(1)

(2)

(3)

(4)

(b6)(b7)

MNA – An example

0

6

0

0

0

001077

000110

1011

00

0111

00

0100111

0000111

5

7

6

4

3

2

1

88

88

433

32

32

1

ES

i

i

i

e

e

e

e

EE

RR

RR

RRR

RG

RG

R

s

MSi

e

C

BYn

0

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