Graph

BASICS OF ELECTRICAL CIRCUITS
EHB 211 E
Circuit Topology & Graphs
Asst. Prof. Onur Ferhanoğlu
1
Circuit Graphs
Graphs retain all interconnection properties, but suppress elements
circuit
(di)graph
• Current direction is preserved
w.r.t the circuit:
current direction points from
+v sign towards –v sign
• No need to mark voltage signs
in digraphs
• Circuit element is suppressed
(deleted)
• power delivered to element
P(t) = v(t)i(t)
Asst. Prof. Onur Ferhanoğlu
circuit
(di)graph
• Only 2 voltages are independent in a 3 terminal element
-> KVL: v1-3 + v3-2 + v2-1 = 0
• Similarly, 2 independent current exists
• -> KCL: i1+i2+i3 = 0
• Therefore, only 2 branches exist in the graph, given that
3 is the datum (reference) node.
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Circuit Graphs
n-terminal element and corresponding graph.
The graphs has n-1 branches
circuit
(di)graph
Power delivered to the n-terminal element:
Asst. Prof. Onur Ferhanoğlu
Graphs/ BASICS OF ELECTRICAL CIRCUITS
3
Circuit Graphs
circuit
(di)graph
Exercise
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
4
Circuit Graphs – two ports
Two port is a circuit (element) with two pairs of accessible terminals:
Example: transformers, hi-fi’s
• KCL: -> i1 = i1` & i2 = i2`
• Power delivered: P= v1(t)i1(t) + v2(t)i2(t)
• The graph of a two port(4 terminal) circuit
contains 2 branches, but the graph of a 1
port 4 terminal circuit contains 3 branches
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
graph
5
Circuit Graphs – multi-ports
Three-winding transformer
Asst. Prof. Onur Ferhanoğlu
circuit
Graphs/ BASICS OF ELECTRICAL CIRCUITS
graph
6
Circuit Graphs – grounded 2-ports
If a common connection exists between nodes 1` and 2`, the circuit is called grounded 2-port,
is equivalent to a 3-terminal circuit
Asst. Prof. Onur Ferhanoğlu
Graphs/ BASICS OF ELECTRICAL CIRCUITS
7
Cut Sets and KCL
Cut set (ξ) is an important graph-theoretical concept:
ξ of a Gaussian surface is called a cut set if
• Removal of all branches of the cut set results in an unconnected graph
• If you leave 1 branch within the cut set, the digraph stays connected
Asst. Prof. Onur Ferhanoğlu
Graphs/ BASICS OF ELECTRICAL CIRCUITS
8
Cut Sets and KCL
KCL: the sum of currents within a cut set is 0
Arrow of the cut set is its reference direction
i1 + i2 – i3 = 0
Proof:
node 5: i4 – i2 – i5 = 0 (node 6: i3 = -i5)
i4 -i2 +i3 = 0 (node 4: i4 = -i1)
-i1 –i2 +i3 = 0
Cut set partitions set of nodes into 2 subsets
By writing KCL for each node and adding the result, we obtain the cut-set equation
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
9
Matrix Formulation - KCL
A digraph with 4 nodes and 6 branches
KCL:
Branches: 1 2 3 4 5 6
i1 + i2
– i6 = 0
-i1
-i3 +i4
=0
- i2 +i3 + i5
=0
- i4 – i5 + i6 = 0
Incidence matrix: Aa
n: # of nodes
Rank: # of independent equations
= n-1 -> 4 nodes -> rank: 3
A
Reduced incidence matrix: Ai = 0
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
10
Matrix Formulation - KCL
A digraph with 4 nodes and 6 branches
All distinct cut-sets of the Graph
Cut-set
matrix
rank: n-1 = 3
Reduced:
Asst. Prof. Onur Ferhanoğlu
QR I = 0
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Matrix Formulation - KVL
A digraph with 4 nodes and 6 branches
datum
Branch voltages
v1 = e1 – e2
v2 = e1
- e3
v3 =
-e2 + e3
v4 =
e2
v5 =
e3
v6 =-e1
v1
v2
v3
v4
v5
v6
=
1
1
0
0
0
-1
-1
0
-1
1
0
0
0
-1
1
0
1
0
e1
e2
e3
e4
e5
e6
Matrix form v = Me
v = AT
v = AT e
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Subgraph
Graph
•
•
•
•
Subgraph
V nodes
B branches
Each branch is incident to two nodes
connected
Asst. Prof. Onur Ferhanoğlu
• V` nodes, subset of V
• B` branches, subset of B
• Does not have to be connected
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Loop
Loop is a connected subgraph, in which 2 branches are incident with each node.
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Loop and KVL
7 distinct loops of the graph
Loop matrix
Linearly dependent equations!
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
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Loop and KVL
3 loops are enough to represent all nodes!
BR v = 0
Linearly independent equations!
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
16
Planar graph
A Planar graph is a graph, which can be drawn on a plane such that no two branches intersect at a point,
which is not a node
Examples of non-planar graphs
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
17
Tree
A Tree is a subgraph that is
• Connected
• Contains all the nodes of the graph
• Has no loops
• Tree branches: twigs
• Branches that do not belong to the tree within a graph: links & chords & cotree branches
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
18
Fundamental Theorem of graphs
1)
There is a unique path along the tree between any pairs of nodes
since a tree is connected
2) There are n-1 twigs and l=b-(n-1) links
3) Every twig together with some links define a unique cut set, called fundamental cut set
associated with the twig
4) Every link and the unique path on the tree between its two nodes constitute a unique loop
called the fundamental loop associated with the link
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
19
Fundamental Cut-Sets Associated with a Tree: KCL based on fundamental cut-sets
n = 6, b = 9
-> 5 twigs, 4 links
Each twig defines a unique fundamental cut-set
(Fundamental Theorem of graphs # 3)
1
Ql
Q.i = 0
Q: (n-1)*b matrix: fundamental cut-set matrix
Q = [1n-1Ql]
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
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KVL using twig voltages
Twig voltages
v = QT vt
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
21
Fundamental Loop Matrix associated with a Tree:
n = 6, b = 9
-> 5 twigs, 4 links
5 fundamental cut sets
Each link defines a unique fundamental loop
(Fundamental Theorem of graphs # 4)
Bv = 0
B = [Bt 1]
Fundamental loop matrix
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
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KCL equation using link currents
BQT = 0
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
i = BT il
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Tellegen’s Theorem
Let ik be branch currents and
vk be branch voltages
Conversation of power requires that the total power delivered to each branch
from the rest of the circuit sums up to 0
Example 2:
i1 =
i2 =
i3 =
i4 = i5 = i6 =
------------------------------v1 = v2 = v3 =
v4 = v5 = v6 =
Example 1:
i1 = 1 i2 = 2 i3 = 3
i4 = -3 i5 = -1 i6 = 4
------------------------------v1 = 2 v2 = 1 v3 = 1
v4 = 6 v5 = 5 v6 = 4
2*1 + 1*2 + 1*3 - 6*3 – 5*1 + 4*4 = 0
Asst. Prof. Onur Ferhanoğlu
Graphs / BASICS OF ELECTRICAL CIRCUITS
24

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