Formulation of Circuit Equations

Lecture 2

Alessandra Nardi

Thanks to Prof. Sangiovanni-Vincentelli and Prof. Newton

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 ….

SPICE historyProf. 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)

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

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

Formulation of Circuit Equations

Circuit withB branches

N nodes

Simulator

Set ofequations

Set ofunknowns

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

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

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

+

_

v1

i1

Two-port

i1

+

_

v2

i2

i2

Branch Constitutive Equations (BCE)

Ideal elementsElement Branch Eqn

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 = ?

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 eb 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.

Equation Formulation - KCL

0

1 2

R1G2v3

R3

R4Is5

0

0

11100

00111

5

4

3

2

1

i

i

i

i

i

A i = 0

Kirchhoff’s Current Law (KCL)

N equations

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

v

v - AT e = 0

Kirchhoff’s Voltage Law (KVL)

B equations

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 + i = is B equations

Equation FormulationNode-Branch Incidence Matrix

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

PROPERTIES•A is unimodular•2 nonzero entries in each column

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

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

00N+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

ProblemSophisticated programming techniques and datastructures are required for time and memoryefficiency

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)

Yne=ins

N eqnsN unknowns

N = # nodesNodal 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

R

Nodal Analysis

• Example shows NA may be derived from STA

• Better: 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”

ik

N+ N-

+ -Ek

Spice input format: ESk 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 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

MNA – CCCS and CCVS “Stamp”

MNA – An example

Step 1: Write KCLi1 + i2 + i3 = 0 (1)-i3 + i4 - i5 - i6 = 0 (2)i6 + i8 = 0 (3)i7 – i8 = 0 (4)

0

1 2

G2v3

R4Is5R1

ES6- +

R8

3

E7v3

- +4

MNA – An exampleStep 2: Use branch equations to eliminate as many branch currents as possible1/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 equationsv6 = ES6 (b6)v7 – E7·v3 = 0

(b7)

MNA – An exampleStep 4: Use KVL to eliminate branch voltages from previous equations1/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)

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 and principle minors may also be singular.