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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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 2
Graph and digraph♦ Consists of branches and nodes♦ Describes the interconnection of the elements
Graph
Digraph— arrowsindicate directions ofcurrents and voltages’polarities
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 3
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 4
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 5
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 6
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 7
KCLThe following are all KCL equationsfor the circuit below:
–Ia + Ib + Id = 0Ic + Id + Ib = 0Ic + Id + Ie = 0
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 8
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 9
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 10
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 11
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 12
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 13
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!
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 14
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)
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 15
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 16
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 ]
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 17
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 ]
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 18
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 19
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 20
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!
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 21
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 22
Another example
CLEVER METHOD:
We define unknowns such that the 1A sourceand 2A source are exactly the unknowns. Then,we save two equations!
So, we have 0 equation with 0 unknown.
Usual mesh assignment:
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 23
Question
How to make the clever method a general methodsuitable for all cases?
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 24
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 –
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 25
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.
So, we have 0 equation with 0 unknown.
+ V1 – + V2 –
+ V1 –
+ V2 –
+ V3 –
+ V3 –
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 26
Same question
How to make the clever method a general methodsuitable for all cases?
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 27
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?
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 28
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 29
Standard tree
number of nodes n = 4number of branches b = 5number of tree branches t = n–1 = 3
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 30
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 31
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:
⇒
⇒
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 32
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 33
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!
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 34
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.
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 35
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
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Prof. C.K. Tse: Graph Theory &Systematic Analysis 36
Conclusion
Graph theoryTake advantage of topology
Cutset-voltage approachAim to find all tree voltages initially
Loop-current approachAim to find all cotree currents initially