Formula Sheets: Graph Theory | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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Net w ork Theory: Graph Theory F orm ula Sheet for
Electrical GA TE
Net w ork G raph Basics
• Net w ork Graph : Directed graph represen ting a circuit, with no des (v ertices) as
junctions and branc hes (edges) as circuit elemen ts (resistors, sources, etc.).
• T erms :
– No de : Junction p oin t in the circuit.
– Branc h : Circuit elemen t connecting t w o no des.
– T ree : Connected subgraph with no lo ops, con taining all no des.
– Co-tree : Branc hes not in the tree (links or c hords).
– Lo op : Closed path in the graph.
• Relations : F or a graph with n no des and b branc hes:
Num b er of tree branc hes = n-1
Num b er of links (co-tree branc h es) = b-(n-1)
Num b er of fundamen tal lo ops = b-n+1
Incidence Matrix
• No de Incidence Matrix (A
a
) : Ro ws = no des, columns = branc hes.
A
a
[i,j] =
?
?
?
?
?
+1 if branc h j is directed a w a y from no de i
-1 if branc h j is directed to w ard no de i
0 if branc h j is not inciden t on no de i
• Reduced Incidence Matrix (A ) : Delete one ro w (reference no de); size (n-1)×b .
• K CL in Matrix F orm : AI
b
= 0 , where I
b
is the branc h curren t v ector.
Tie-Set Matrix (F undamen tal Lo op Matrix)
• Definition : Represen ts fundamen tal lo ops formed b y adding one link to the tree.
• Tie-Set Matrix (B ) : Ro ws = fundamen tal lo ops, columns = branc hes.
B[i,j] =
?
?
?
?
?
+1 if branc h j is in lo op i with same direct ion
-1 if branc h j is in lo op i with opp osite direc tion
0 if branc h j is not in lo op i
• KVL in Matrix F orm : BV
b
= 0 , where V
b
is the branc h v oltage v ector.
• Relation with Incidence Matrix : BA
T
= 0 (orthogonalit y).
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Net w ork Theory: Graph Theory F orm ula Sheet for
Electrical GA TE
Net w ork G raph Basics
• Net w ork Graph : Directed graph represen ting a circuit, with no des (v ertices) as
junctions and branc hes (edges) as circuit elemen ts (resistors, sources, etc.).
• T erms :
– No de : Junction p oin t in the circuit.
– Branc h : Circuit elemen t connecting t w o no des.
– T ree : Connected subgraph with no lo ops, con taining all no des.
– Co-tree : Branc hes not in the tree (links or c hords).
– Lo op : Closed path in the graph.
• Relations : F or a graph with n no des and b branc hes:
Num b er of tree branc hes = n-1
Num b er of links (co-tree branc h es) = b-(n-1)
Num b er of fundamen tal lo ops = b-n+1
Incidence Matrix
• No de Incidence Matrix (A
a
) : Ro ws = no des, columns = branc hes.
A
a
[i,j] =
?
?
?
?
?
+1 if branc h j is directed a w a y from no de i
-1 if branc h j is directed to w ard no de i
0 if branc h j is not inciden t on no de i
• Reduced Incidence Matrix (A ) : Delete one ro w (reference no de); size (n-1)×b .
• K CL in Matrix F orm : AI
b
= 0 , where I
b
is the branc h curren t v ector.
Tie-Set Matrix (F undamen tal Lo op Matrix)
• Definition : Represen ts fundamen tal lo ops formed b y adding one link to the tree.
• Tie-Set Matrix (B ) : Ro ws = fundamen tal lo ops, columns = branc hes.
B[i,j] =
?
?
?
?
?
+1 if branc h j is in lo op i with same direct ion
-1 if branc h j is in lo op i with opp osite direc tion
0 if branc h j is not in lo op i
• KVL in Matrix F orm : BV
b
= 0 , where V
b
is the branc h v oltage v ector.
• Relation with Incidence Matrix : BA
T
= 0 (orthogonalit y).
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Cut-Set Matrix
• Definition : Represen ts fundamen tal cut-sets, eac h separating the graph in to t w o
parts b y cutting one tree branc h.
• Cut-Set Matrix (Q ) : Ro ws = cut-sets, columns = branc hes.
Q[i,j] =
?
?
?
?
?
+1 if branc h j is in cut-set i with same direction
-1 if branc h j is in cut-set i with opp osite direction
0 if branc h j is not in cut-set i
• K CL in Matrix F orm : QI
b
= 0 .
• Relation with Incidence Matrix : QA
T
= 0 .
Applications in Circuit Analysis
• No dal Analysis : Use reduced incidence matrix to write K CL equations.
AY
b
A
T
V
n
= AI
s
where Y
b
is the branc h admittance matrix, V
n
is the no de v oltage v ector, I
s
is the
source curren t v ector.
• Mesh/Lo op Analysis : Use tie-set matrix to write KVL equations.
BZ
b
B
T
I
m
= BV
s
where Z
b
is the b ranc h imp edance matrix, I
m
is the mesh curren t v ector, V
s
is the
source v oltage v ector.
• Cut-Set Analysis : Use cut-set matrix for K CL-based analysis.
Key Notes
• Use consisten t orien tation for branc h directions in matrices.
• Num b er of fundamen tal lo ops = n um b er of links, n um b er of fundamen tal cut-sets =
n um b er of tree branc hes.
• F or GA TE, fo cus on tie-set and cut-set matrices for lo op and no dal analysis.
• Ensure prop er selection of tree to simplify matrix form ulation.
• Matrices are used to systematize K CL and KVL for complex circuits.
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