computer-aided analysis of electronic circuits” course notes 3

Faculty of

Electronics, Telecommunications and Information

Technologies

Gheorghe Asachi Technical University of

Iasi

www.etti.tuiasi.rotuiasiC www.study.tuiasi.ro www.learning.tuiasi.ro

” Computer-Aided Analysis

of Electronic Circuits”

Course Notes 3

Lecturer: Dan Burdia, PhD

Bachelor: Telecommunication Technologies and Systems

Year of Study: 2

Contents

• I. Introduction in Computer-Aided Analysis.

•

• II. Computer Circuit Models of Electronic Devices and Components

• III. Network Topology: The Key to Computer Formulation of the

Kirchhoff Laws

• IV. Nodal Linear Network Analysis: Algorithms and Computational

Methods

• V. Nodal Nonlinear Resistive Network Analysis: Algorithms and

Computational Methods

2

Chapter III - Network Topology: The Key to Computer Formulation of the Kirchhoff Laws

• 3.1 Basic concepts in network topology

• 3.2 Incident matrix

• 3.3 Loops matrix

• 3.4 Cutset matrix

• 3.5 Basic relations between branch variables

• 3.6 Computer-aided formulation of A, B and D matrices

3

3.1 Basic concepts in network topology

4

Circuit schematic Directed graph Undirected graph

Graph - consists of a set of nodes connected by branches.

Graph types:

- Directed graph

- Undirected graph

- Connected graph

- Unconected graph

Connected graph – there is a path between any

two nodes from graph

Unconnected graph – a graph with one or more

isolated (separated) nodes.

Unconnected graph

• Loop

▫ A subgraph Gs of a graph Gn which fulfills the following two conditions:

� Gs is connected (connex)

� At each node in Gs there are exactly two branhes of the Gs

Example: 1,2,6; 1,3,4,5,6; 3,4,7,6,2; Not loop: 3,4,5,2,7

• Cut-set▫ A set of branches which fulfills the following two conditions:

� If the set is removed from graph then the graph becomes unconnected

� Re-inserting any branch from set the graph becomes again connected.

Example: 1,2,3; 3,5,7; 2,5,6; Not cutset: 3,2,5,7

• Tree

▫ A subgraph Gs of a graph Gn which fulfills the following three conditions:

� Gs is connected

� Gs contains all nodes from Gn

� Gn does not contain loops.

Example: 3,2,5,6; 3,4,6,7; Not tree: 1,2,6,5; 3,2,4,5,7

3.1 Basic concepts in network topology (cont)

Complete incident matrix (Aa). For a directed graph Gd having n nodes and b

branches, Aa is a n x b matrix, Aa = [aij], where:

- aij = 1, if branch j is incident to node i, sense goes out from node

- aij = -1, if branch j is incident to node i, sense goes in to node

- aij = 0, if branch j is not incident to node IExample:

3.2 Incident matrix

Reduced incident matrix (A) – is obtained from Aa removing a row (usually the

row of ground node)

First Kirchhoff Law for all nodes:

3.2 Incident matrix (cont.)

Theorem 3.1. For a directed graph Gd the rows of reduced incident matrix A are

linear independent.

Corolar. The maximum set of independent First Kirchhoff Law equations are

written as: 0A i

Theorem 3.2. Given A the reduced incident matrix for a directed graph Gd having

n nodes. Then n-1 columns from A are linear independent if and only if the

branches corresponding to those columns form a tree in Gd.

Corolar. Let consider A partitioned as:

where AT and AL have the columns corresponding to the tree branches and co-tree

branches, respectively. Then det(AT) ≠ 0.

T LA A A M

Example:

3.3 Loops matrix

Definition. Complete loops matrix Ba

For a directed graph Gd with n nodes, b branches and nb directed loops,

Ba = [bij]nb x b, where:

bij= +1 if branch j belongs to the loop i, the same orientation

bij= -1 if branch j belongs to the loop i, the opposite orientation

bij= 0 if branch j does not belong to the loop i

Kirchhoff II Law for all possible loops (not all are independent):

Base loops matrix Bb: a submatrix of Ba having maximum independent rows from Ba

Bb have b-n+1 lines.

How we can select a set of b-n+1 independent rows (loops)?

Systematic approach: by determining the fundamental loops

Fundamental loop: contains only one branch from co-tree and the other

branches are from the tree. The loop orientation is the same to branch from co-tree.

Fundamental loops matrix B can be partitioned as B = [BT | 1]

3.3 Loops matrix (cont.)

Example: Fundamental loops matrix B for a graph. Tree = a,d,e,f, i.

Theorem 3.3. If the columns of the matrices Aa and Ba are arranged in the same

order of branches, then for any i and j:

[row i from Ba] x [row j from Aa]t = 0 .

Corolar: The following relations are true

3.3 Loops matrix (cont.)

Theorem 3.4. For a directed graph Gd having n nodes and b branches the maximum

number of independent rows from Ba is b-n+1.

Demonstration: consider the tree T. Then Ba can be partitioned as:

Starting from it is easy to show that B2 = B2LB. This means only the rows of

B are the independent rows from Ba. 0t

aB A

The maximum set of Kirchhoff II Law equations can be written as B·v=0

3.4 Cutsets matrix

Definition. Complete cutset matrix Da

For a directed graph Gd with n nodes, b branches and ns directed cutsets,

Da = [dij]ns x b, where:

dij= +1 if branch j belongs to the cutset i, the same orientation

dij= -1 if branch j belongs to the cutset i, the opposite orientation

dij= 0 if branch j does not belong to the cutset i

Generalized Kirchhoff I Law for all possible cutsets (not all are independent):

Base cutsets matrix Db: a submatrix of Da having maximum independent rows from Da

Db have n-1 lines.

How we can select a set of n-1 independent rows (cutsets)?

Systematic approach: by determining the fundamental cutsets

Fundamental cutset: contains only one branch from tree and the other branches

are from the co-tree. The cutset orientation is the same to branch from the tree.

Fundamental cutsets matrix D can be partitioned as D = [1 | DL].

3.4 Cutsets matrix (cont.)

Example 1. Complete cutset matrix Da

Example 2. Fundamental cutset matrix D for a graph. Tree T = a,d,e,f,i.

Theorem 3.5. If the columns of the matrices Ba and Da are arranged in the same order of

branches, then for any i and j:

[row i from Ba] x [row j from Da]t = 0 .

Corolar: The following relations are true

3.4 Cutsets matrix (cont.)

Theorem 3.6. For a directed graph Gd having n nodes and b branches the maximum

number of independent rows from Da is n-1.

Demonstration: consider the tree T. Then Da can be partitioned as:

2 2 2

1 L

a

T L

D D

D

D D D

M

M

0t

aD B Starting from it is easy to show that D2 = D2TD. This means only the rows of

D are the independent rows from Da.

The maximum set of generalized Kirchhoff I Law equations can be written as D·i = 0

3.5 Basic relations between branch variables

Given Gd with n nodes and b branches, let partition the matrices A, B and D as:

1 1T L T LA A A B B D D M M M 1b n 1n

Branch voltages and branch currents:

T T

L L

v iv i

v i

0tB D 1

1 0 0t

T T L

t

L

B B D

D

M L=> =>t

T LB D or t

L TD B

0B v 1 0T

T

L

vB

v

M=> L T Tv B v => 0T T LB v v =>

0D i => 1 0 0T

L T L L

L

iD i D i

i

M =>

T L Li D i

3.5 Basic relations between branch variables (cont.)

11

tttT T tT L

L T L L

L L

i BB ii i B i B i

i i

M

11

tT T t

T L T Tt t

L L T L

v vv v D v D v

v D v D

M

Branch currents as a function of co-tree branch currents

Branch voltages as a function of tree branch voltages

Nodal voltages vector:

1

2

, 1

n

n

n

n n

v

vv

v

M

Relation between branch voltages and nodal voltages:t

nv A v

3.6 Computer-aided generation of the matrices A, B and D

Generation of incident matrix A

- Assign consecutive numbers to nodes. 0 is for reference node (ground)

- Assign consecutive numbers to network branches

- If branch k is between nodes i and j, this can be stored as a triplet (k, i, j). Thus:

1ika 1jka

Generation of fundamental matrices B and D

- the network tree must be determined firstly

There are two stages:

1 - Determination of the network tree T with a preferential order of branches to be

included in the tree. Such tree is named normal tree.

2 - Generation of the matrices D and B

3.6.1 Network tree determination

•The network tree can be determined starting from A matrix

•The columns of A matrix are rearranged in the preferential order of branches to be

included in the tree:

Independent voltage sources

Controlled voltage sources

Capacitors

Resistors

Inductors

Controlled current sources

Independent current sources

9 1 6 2 5 8 10 3 4 7

1

2

1

V C C R R R L I J J

A

n

M

Example:

• In the matrix A first independent n-1

columns are determined. This is possible

by transforming A matrix into an echelon

form matrix using elementary row

operations:

• row interchanging

• multiply a row with a non-zero value

• multiply a row with a non-zero value

and add to another row

1 2 3

1 1 x x x x x x x

2 0 0 1 x x x x x

3 0 0 0 1 x x x x

0 0 0 0 0 0 1 x

0 0 0 0 0 0 0 0

kc c c c

Q

k

L L

L L

L L

M M M M M M M M

L L

L L

Echelon form matrix:

Network tree determination – cont.

Algorithm to transform the matrix A into echelon form matrix

Let As a submatrix of A. Initial As=A

• Step 1: the first nonzero column is searched in As. Name c this column.

• Step 2: In column c, first non-zero value is searched from up to down, for

example in the row r. If r=1 go to step 4.

• Step 3: Within As, rows 1 and r are interchanged.

• Step 4: If element (1,c) is -1 the row 1 from As is multiplied by -1.

• Step 5: In column c all non-zero elements below first row from As are reduced to

zero.

• Step 6: New As matrix is defined. It consist of rows below first row from old As

and columns situated at the right side of column c. If new As doesn’t exist,

algorithm is finished, else go to step 1.

•Independent columns in A are the

columns of first element of 1 from each

row within echelon form matrix

• Branches corresponding to these

columns form the network tree.

Network tree determination – Example

3 2 4 8 1 9 10 5 6 7

2 1

1 5 5

1 0 1 1 0 1 0 0 1 0 0

2 1 1 0 0 1 0 0 0 0 0

3 0 0 0 0 0 0 0 0 1 1

4 0 0 0 1 0 1 0 1 1 0

5 1 0 1 0 0 1 1 0 0 0

L L

L L L

V C C C R R R L L I

A

Matrix A with columns arranged in the preferential order of branches:

Network tree determination – Example (cont.)

Transform matrix A into echelon form using elementary row operations:

Normal tree: V3, C2, C8, R9, L6

3.6.2 Generating matrices B and D

Assume the network tree T is determined and matrices A, B and D are partitioned as

1

1

T L

T

L

A A A

B B

D D

M

M

M

1

1

b n

n

n – number of nodes

b – number of branches

Matrices B and D can be obtained from matrix A.

0 0 01

tt tT

T L T T L

BA B A A A B A

M

1tT T LB A A

t

T LB D

1L T LD A A

1 1 1 11 L T T T L T T L TD D A A A A A A A A A M M M

1 1tT LB B D M M

Disadvantage: calculus of1TA

Another method: based on elementary operations

3.6.2 Generating matrices B and D (cont.)

Alternative method can be deduced using elementary matrices.

Definition: An elementary matrix is a matrix obtained from unit matrix after an

elementary operation.

Theorem: performing an elementary operation on a matrix Q is equivalent with

the product εQ, where ε is the elementary matrix corresponding to that operation.

Example:

( 2) 2 1 11

3 2 1 4 1 6 5 2

2 4 2 1 2 4 2 1

0 5 3 2 0 5 3 2

L L LQ Q

1

1 2 0 1 6 5 2

0 1 0 2 4 2 1

0 0 1 0 5 3 2

Q Q

If Q is a square and non-singular matrix, then there is an sequence of elementary

operation to reduce Q to a unity matrix. AT is a square and non-singular matrix!

3 2 1... 1m Q 13 2 1...mQ

13 2 1...T mA


13 2 1...T mA

13 2 1

3 2 1 3 2 1

...

... ... 1

T m T L

m T m L L

D A A A A

A A D

Thus, D can be obtained from A by elementary operations until sub-matrix AT is reduced

to a unity matrix.

Algorithm to reduce a square and non-singular matrix Q to a unity matrix:

1) At first Q is reduced to echelon form matrix Qes

2) Non-zero elements above diagonal are reduced to zero using the ones from

diagonal


Example 1: Determination of matrices D and B starting from following incident matrix

1 1 1 0 1 0 0 1

2 1 0 0 1 1 0 0

3 0 0 0 1 0 1 0

4 0 0 1 0 1 1 1

T L

a b c d e f g

A A A

M

Step 1: Reduce A to an echelon matrix

1 2 2

1 1 0 0 0 0 1

0 1 0 1 1 0 1

0 0 0 1 0 1 0

0 0 1 0 1 1 1

L L L

3 4

3( 1)4( 1)

1 1 0 0 0 0 1

0 1 0 1 1 0 1

0 0 1 0 1 1 1

0 0 0 1 0 1 0

L L

LL

Step 2: Reduce to zeros all non-zero elements above diagonal

( 1) 4 2 2

( 1) 2 1 1

1 0 0 0 1 1 0

0 1 0 0 1 1 11

0 0 1 0 1 1 1

0 0 0 1 0 1 0

L L LLL L L

a b c d e f g

D D

M


1 1 1 0 1 0 0

1 1 1 1 1 1 0 1 0

0 1 1 0 0 0 1

tT L

a b c d e f g

B B D

M M

Loops matrix B

Example 2

3 2 4 8 1 9 10 5 6 7

1 0 1 1 0 1 0 0 1 0 0

2 1 1 0 0 1 0 0 0 0 0

3 0 0 0 0 0 0 0 0 1 1

4 0 0 0 1 0 1 0 1 1 0

5 1 0 1 0 0 1 1 0 0 0

V C C C R R R L L I

A


Columns of A are rearranged in form A=[AT, AL ]

3 2 8 9 6 4 1 10 5 7

1 0 1 0 0 0 1 1 0 1 0

2 1 1 0 0 0 0 1 0 0 0

3 0 0 0 0 1 0 0 0 0 1

4 0 0 1 1 1 0 0 0 1 0

5 1 0 0 1 0 1 0 1 0 0

T L

V C C R L C R R L I

A A A

M

The algorithm to reduce AT to a unity matrix is applied:

3 2 8 9 6 4 1 10 5 7

1 1 0 0 0 0 1 0 0 0

0 1 0 0 0 1 1 0 1 0

0 0 1 1 1 0 0 0 1 0

0 0 0 1 0 0 0 1 1 0

0 0 0 0 1 0 0 0 0 1

es

V C C R L C R R L I

A

Echelon form matrix

3 2 8 9 6 4 1 10 5 7

5 3 3

( 1) 4 3 3

( 1) 2 1 1

1 0 0 0 0 1 0 0 1 0

0 1 0 0 0 1 1 0 1 01

0 0 1 0 0 0 0 1 0 1

0 0 0 1 0 0 0 1 1 0

0 0 0 0 1 0 0 0 0 1

L L L

L L L

L L L

L

V C C R L C R R L I

D D

M

Matrix D

3 2 8 9 6 4 1 10 5 7

1 1 0 0 0 1 0 0 0 0

0 1 0 0 0 0 1 0 0 01 1

0 0 1 1 0 0 0 1 0 0

1 1 0 1 0 0 0 0 1 0

0 0 1 0 1 0 0 0 0 1

t

T L

V C C R L C R R L I

B B D

M M


Loops matrix: can be written directly from D, taking into accountt

T LB D

computer-aided analysis of electronic circuits” course notes 3

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