Graph theory and systematic analysis Electronic Circuits 1

Electronic Circuits 1

Graph theory and systematic analysis

Contents:

• Graph theory

• Tree and cotree

• Basic cutsets and loops

• Independent Kirchhoff’s law equations

• Systematic analysis of resistive circuits

• Cutset-voltage method

• Loop-current method

Prof. C.K. Tse: Graph Theory &

Systematic Analysis 1

♦

♦

Graph and digraph

Consists of branches and nodes

Describes the interconnection of the elements

Graph


Systematic Analysis

Digraph— arrows indicate directions of currents and voltages’ polarities

2

Sign convention

♦

Stick to the following sign convention

♦

♦

Current direction — same as arrow direction

Voltage polarity — arrow goes from + to – through the element

I

+ V –



Loop

♦

A loop is a set of branches of a graph forming a closed path.

♦

For example,

♦ branches a, c, d

♦ branches a, b, e, c



Cutset

♦

A cutset is a set of branches of a graph, which upon removal will cause the graph to separate into two disconnected sub-graphs.

Examples: branches f, b, d, c

SPECIAL CASE

Branches emerging from a node form a cutset always a cutset



Kirchhoff’s laws again

KVL — same as before.

KCL — more generally stated in terms of cutset with appropriately chosen directions

Usually the cutset separates the graph into two subgraphs. We may say that the sum of currents going from one sub-graph to the other is zero.



KCL

The following are all KCL equations for the circuit below:

–I a

+ I b

+ I d

= 0

I

I c

+ c

+ I

I d d

+

+ I

I b

= 0 e

= 0



Problem: Find

I y

I w


Systematic Analysis

Usual way:

Find I z

Then find I

Then find I x w

Then we get I y

Alternative way:

Using KCL for an appropriate cutset, the

I problem is as simple as y

+ 5 + 3 = 0!

8

Tree and co-tree

A tree is a set of branches of a graph which contains no loop. Moreover, including one more branch to this set will create a loop.

Thus, a tree is a maximal set of branches that contains no loop.

After a tree is chosen, the remaining branches form a co-tree .

— tree

…. co-tree



Basic relations

Let n = number of nodes b = number of branches t = number of tree branches l = number of co-tree branches

We have, for all planar graphs, t = n – 1 l = b – t = b – n + 1



Basic cutsets

A basic cutset is a cutset containing only one tree branch.

So, there are t basic cutsets in a graph.

In this example, the basic cutsets are

{ 1 , 3, 6 }

{ 2 , 3, 5 }

{ 4 , 5, 6 } tree branches

The importance of basic cutsets is the formulation of independent KCL equations:



Basic loops

A basic loop is a loop containing only one co-tree branch.

So, there are t basic cutsets in a graph.

In this example, the basic cutsets are

{ 1, 2, 3 }

{ 2, 4, 5 }

{ 1, 4, 6 } co-tree branches

The importance of basic loops is the formulation of independent KVL equations:



Independent KCL/KVL equations

A different choice of tree gives a different set of basic cutsets and basic loops.

The set of independent KCL and KVL equations found is not unique.

But any set of independent KCL and KVL equations gives essentially the same information about the circuit. So, it doesn’t matter which tree is chosen.

Once a tree is chosen, a set of independent KCL and KVL equations is found.

Any other KCL or KVL equation is derivable from the independent set. That means, we DON’T NEED to find more than t KCL or b–t KVL equations, since anything more than the basic set is redundant and a waste of effort!



Matrix representations

There are three fundamental matrices representing the graph of a given circuit:

1.

Node-incidence matrix (A-matrix)

2.

Basic cutset matrix (Q-matrix)

3.

Basic loop matrix (B-matrix)

They are very useful in computer-aided systematic analysis.



Node-incidence matrix (A-matrix)

The A-matrix describes the way a circuit is connected. It is very important in computer simulation.

The columns in a A-matrix correspond to the branches; and the rows correspond to the nodes.



Basic cutset matrix (Q-matrix)

The Q-matrix describes the way the basic cutset is chosen.

Each column corresponds to a branch

( b columns).

Each row corresponds to a basic cutset

( t rows).

Construction

For each row:

Put a “+1” in the entry corresponding to the cutset tree branch.

Put a “0” in the entry corresponding to other tree branches.

Put a “+1” or “–1” in the entry corresponding to each cutset co-tree branch; “+” if it is consistent with the tree branch direction and “–” otherwise.

Q = [ 1 | Q

1

]



Basic loop matrix (B-matrix)

The B-matrix describes the way the basic loop is chosen.

Each column corresponds to a branch

( b columns).

Each row corresponds to a basic loop

( b – t rows).

Construction

For each row:

Put a “+1” in the entry corresponding to the loop co-tree branch.

Put a “0” in the entry corresponding to other co-tree branches.

Put a “+1” or “–1” in the entry corresponding to each loop tree branch;

“+” if it is consistent with the co-tree branch direction and “–” otherwise.

B = [ B

1

| 1 ]



Relationship between

Q

and

B

Q = [ 1 | Q

1

] B = [ B

1

| 1 ]

It is always true that Q

1

= – B

1

T or B

1

= – Q

1

T

Thus, once we have Q , we know B , and vice versa.



Applications

The basic cutset and loop matrices will be used to formulate independent Kirchhoff’s law equations. This will give much more efficient solution to circuit analysis problems.

Mesh —enhanced— General loop analysis

Nodal —enhanced— General cutset analysis



Recall: mesh analysis

Mesh analysis

— good for circuits without current sources

Problem occurs when circuits have a current source: WASTE OF EFFORT!

WHY?

The unknowns are actually partially known!



Redundancy in mesh analysis

USUAL MESH ANALYSIS:

Obviously if we define the unknowns according to the usual mesh-analysis.

We have 2 equations with 2 unknowns.

This is UNNECESSARY because the current source actually gives the current values indirectly! I

1

– I

2

= 1 A.

CLEVER METHOD:

We define unknowns such that the 1A source is exactly one of the unknowns. Then, we save an equation!

So, we have 1 equation with 1 unknown.



Another example

Usual mesh assignment:

CLEVER METHOD:

We define unknowns such that the 1A source and 2A source are exactly the unknowns. Then, we save two equations!




Question

How to make the clever method a general method suitable for all cases?



Redundancy in nodal analysis

USUAL NODAL ANALYSIS:

Obviously if we define the unknowns according to the usual nodal analysis, V

1

, V

2 and V

3 we have 3 equations with 3 unknowns.

This is UNNECESSARY because the voltage source actually gives the voltage values indirectly! V

1

– V

2

= 2 V.

CLEVER METHOD:

+

V

–

1

+

V

–

2

We define unknowns such that the 2V source is exactly one of the unknowns. Then, we save an equation! Here, we use branch voltages.

So, we have 2 (cutset) equations with 2 unknowns.


Systematic Analysis

+ V

1

–

+

V

–

2

+

V

–

3

+

V

–

3

24

Another example

USUAL NODAL ANALYSIS: +

V

–

1

+

V

–

2

+

V

–

3

CLEVER METHOD:

We define unknowns such that the sources overlap with unknown branches. Then, we save three equations! Here, we use branch voltages.



Systematic Analysis

+ V

1

– + V

2

– +

V

–

3

25

Same question

How to make the clever method a general method suitable for all cases?



Key to systematic methods

Graph theory

•Tree / basic cutset KCL equations

•Co-tree / basic loop KVL equations

The first step is define an appropriate tree!

Hint: where should we put all the voltage sources?



Standard tree

Take branches into the tree according to the following priority:

All voltage-source branches

All resistor branches that do not close a path

The remaining all go to the co-tree.

The co-tree will have all the current sources.



Standard tree

number of nodes n = 4 number of branches b = 5 number of tree branches t = n–1 = 3



Two systematic approaches

Once the tree is chosen, we have two possible approaches to solve the problem:

1.

Cutset-voltage approach (c.f. nodal)

Unknowns are tree voltages

Set up KCL equations based on basic cutsets

2.

Loop-current approach (c.f. mesh)

Unknowns are co-tree (link) currents

Set up KVL equations based on basic loops



+

V

–

1

2S

1

1V

– +

Cutset-voltage approach

1

2A

4

2S

1S

3

Step 1:

Start with the digraph. Choose a tree. Define unknowns as the tree voltages. Label all voltages.

Step 2:

Write the KCL equations for each basic cutset

(except those corresponding to voltage sources)

1S

2

+

V

–

2

5

2S

+

–

Cutset 1:

Cutset 2:

⇒

3V

2

⇒



7V

+

–

Loop-voltage approach

1 Ω

7A

3

2 Ω 5

2

4

2 Ω

3 Ω

1 Ω

1

Step 1:

Start with the digraph. Choose a tree. Define unknowns as the co-tree currents. Label all currents.

Step 2:

Write the KVL equations for each basic loop

(except those corresponding to current sources)

Loop 1:

Loop 2:



Choice of method

Cutset-voltage method: Loop-current method:

Equations to be solved

= t – (number of voltage sources)

= n – 1 – (number of voltage sources)

Equations to be solved

= b – t – (number of current sources)

= b – n + 1 – (number of current sources)

CHOOSE THE SIMPLEST!



Question!!

So far, we have only focused on finding

EITHER the tree voltages

OR the co-tree currents

How about other branch currents and voltages?

Can you verify the following:

Once we know either the tree voltages or the co-tree currents, we can derive everything else in the circuit.



Sherlock Holmes’ search

Cutset-voltage method:

Tree: Voltage sources

Resistors

Co-tree: Resistors

Current sources voltage

?

?

KVL B-loop

KVL B-loop current

?

?

?

KCL B-cutset

Ohm’s law

Ohm’s law

Loop-current method:

Tree: Voltage sources

Resistors

Co-tree: Resistors

Current sources voltage

?

?

?

?

Ohm’s law

Ohm’s law

KVL B-loop current

?

?

KCL B-cutset

KCL B-cutset



Conclusion

Graph theory

Take advantage of topology

Cutset-voltage approach

Aim to find all tree voltages initially

Loop-current approach

Aim to find all cotree currents initially



Graph theory and systematic analysis Electronic Circuits 1

Graph theory and systematic analysis

Graph and digraph

Sign convention

Loop

Cutset

Kirchhoff’s laws again

KCL

Problem: Find

Tree and co-tree

Basic relations

Basic cutsets

Basic loops

Independent KCL/KVL equations

Matrix representations

Node-incidence matrix (A-matrix)

Basic cutset matrix (Q-matrix)

Basic loop matrix (B-matrix)

Relationship between

and

Applications

Recall: mesh analysis

Redundancy in mesh analysis

Another example

Question

Redundancy in nodal analysis

Another example

Same question

Key to systematic methods

Standard tree

Standard tree

Two systematic approaches

Cutset-voltage approach

Loop-voltage approach

Choice of method

Question!!

Sherlock Holmes’ search

Conclusion

Graph theory

Cutset-voltage approach

Loop-current approach

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