Graph theory and systematic analysis - EIEcktse/eie201/2.SystematicAnalysis.pdfMesh —enhanced— General loop analysis Nodal —enhanced— General cutset analysis Prof. C.K. Tse:

Prof. C.K. Tse: Graph Theory &Systematic Analysis 1

Electronic Circuits 1

Graph theory andsystematic 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


Graph and digraph♦ Consists of branches and nodes♦ Describes the interconnection of the elements

Graph

Digraph— arrowsindicate directions ofcurrents and voltages’polarities


Sign convention♦ Stick to the following sign convention

♦ Current direction — same as arrow direction♦ Voltage polarity — arrow goes from + to – through the element

+ V –

I


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, whichupon removal will cause the graph to separate intotwo 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 againKVL — 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 thatthe sum of currents going from one sub-graph to the other is zero.


KCLThe following are all KCL equationsfor the circuit below:

–Ia + Ib + Id = 0Ic + Id + Ib = 0Ic + Id + Ie = 0


Problem: Find Iy

Usual way:Find IzThen find IxThen find IwThen we get Iy

Alternative way:Using KCL for anappropriatecutset, theproblem is assimple asIy + 5 + 3 = 0!

Iw


Tree and co-treeA tree is a set of branches of a graph whichcontains no loop. Moreover, including one morebranch to this set will create a loop.

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

After a tree is chosen, the remaining branchesform a co-tree.

— tree…. co-tree


Basic relations

Let

n = number of nodesb = number of branchest = number of tree branchesl = 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 onetree 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

The importance of basic cutsets is theformulation of independent KCL equations:

tree branches


Basic loops

A basic loop is a loop containing only one co-treebranch.

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

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

co-tree branches


Independent KCL/KVL equations

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

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

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

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. Thatmeans, we DON’T NEED to find more than t KCL or b–t KVLequations, since anything more than the basic set is redundant anda waste of effort!


Matrix representations

There are three fundamental matrices representing the graph of a givencircuit:

They are very useful in computer-aided systematic analysis.

1. Node-incidence matrix (A-matrix)2. Basic cutset matrix (Q-matrix)3. Basic loop matrix (B-matrix)


Node-incidence matrix (A-matrix)

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

The columns in a A-matrix correspond to the branches; and the rowscorrespond 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).

ConstructionFor each row:

Put a “+1” in the entry correspondingto the cutset tree branch.Put a “0” in the entry corresponding toother tree branches.Put a “+1” or “–1” in the entrycorresponding to each cutset co-treebranch; “+” if it is consistent with thetree branch direction and “–”otherwise. Q = [ 1 | Q1 ]


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).

ConstructionFor each row:

Put a “+1” in the entry correspondingto the loop co-tree branch.Put a “0” in the entry corresponding toother co-tree branches.Put a “+1” or “–1” in the entrycorresponding to each loop tree branch;“+” if it is consistent with the co-treebranch direction and “–” otherwise.

B = [ B1 | 1 ]


Relationship between Q and B

It is always true that Q1 = – B1T or B1 = – Q1

T

B = [ B1 | 1 ]Q = [ 1 | Q1 ]

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


Applications

The basic cutset and loop matrices will be usedto formulate independent Kirchhoff’s lawequations. This will give much more efficientsolution 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 analysisUSUAL MESH ANALYSIS:

Obviously if we define the unknowns accordingto the usual mesh-analysis.

We have 2 equations with 2 unknowns.

This is UNNECESSARY because the currentsource actually gives the current valuesindirectly! I1 – I 2 = 1 A.

CLEVER METHOD:

We define unknowns such that the 1A source isexactly one of the unknowns. Then, we save anequation!

So, we have 1 equation with 1 unknown.


Another example

CLEVER METHOD:

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


Usual mesh assignment:


Question

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


Redundancy in nodal analysisUSUAL NODAL ANALYSIS:

Obviously if we define the unknowns accordingto the usual nodal analysis, V1, V2 and V 3

we have 3 equations with 3 unknowns.

This is UNNECESSARY because the voltagesource actually gives the voltage valuesindirectly! V1 – V2 = 2 V.

CLEVER METHOD:

We define unknowns such that the 2V source isexactly one of the unknowns. Then, we save anequation! Here, we use branch voltages.

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

+ V1 –+ V2 –

+ V1 –

+ V2 –

+ V3 –

+ V3 –


Another example

USUAL NODAL ANALYSIS:

CLEVER METHOD:

We define unknowns such that the sourcesoverlap with unknown branches. Then, we savethree equations! Here, we use branch voltages.


+ V1 – + V2 –

+ V1 –

+ V2 –

+ V3 –

+ V3 –


Same question

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


Key to systematic methodsGraph 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 treeTake branches into the tree according to thefollowing priority:

All voltage-source branchesAll 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 = 4number of branches b = 5number of tree branches t = n–1 = 3


Two systematic approachesOnce the tree is chosen, we have two possible

approaches to solve the problem:

1. Cutset-voltage approach (c.f. nodal)

2. Loop-current approach (c.f. mesh)

Unknowns are tree voltagesSet up KCL equations based on basic cutsets

Unknowns are co-tree (link) currentsSet up KVL equations based on basic loops


Cutset-voltage approachStep 1:Start with the digraph. Choose a tree. Defineunknowns as the tree voltages. Label all voltages.

– +

+–

2A

1V

3V

2S

2S

1S2S

1S

+ V1 –

+ V2 –

1

2

1

2

3

4

5

Step 2:Write the KCL equations for each basic cutset(except those corresponding to voltagesources)

Cutset 1:Cutset 2:

⇒

⇒


Loop-voltage approachStep 1:Start with the digraph. Choose a tree. Defineunknowns as the co-tree currents. Label all currents.

Step 2:Write the KVL equations for each basic loop(except those corresponding to currentsources)

Loop 1:Loop 2:

+–7V

2Ω

2Ω

1Ω

1Ω

7A

3Ω

1

23

4

5


Choice of method

Cutset-voltage method:

Equations to be solved

= t – (number of voltage sources)

= n – 1 – (number of voltage sources)

Loop-current method:

Equations to be solved

= b – t – (number of current sources)

= b – n + 1 – (number of currentsources)

CHOOSE THE SIMPLEST!


Question!!

So far, we have only focused on finding

EITHER the tree voltagesOR 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 canderive everything else in the circuit.


Sherlock Holmes’ search

Tree: Voltage sources

Resistors

Co-tree: Resistors

Current sources

voltage current

??

???

Tree: Voltage sources

Resistors

Co-tree: Resistors

Current sources

voltage current

??

????

Cutset-voltage method:

Loop-current method:

KVL B-loopKVL B-loop

Ohm’s lawOhm’s law

KCL B-cutsetKCL B-cutsetOhm’s law

Ohm’s law

KCL B-cutset

KVL B-loop


Conclusion

Graph theoryTake advantage of topology

Cutset-voltage approachAim to find all tree voltages initially

Loop-current approachAim to find all cotree currents initially

Documents

Graph theory and systematic analysis - EIEcktse/eie201/2.SystematicAnalysis.pdfMesh —enhanced— General loop analysis Nodal —enhanced— General cutset analysis Prof. C.K. Tse: