Network Equilibrium Equations - Electrical Network Graphs

This is my fifth article on electrical network graphs. The article covers the topic  network equilibrium equations. Network equilibrium equations completely determine the state of the network at any moment.

Key Concept

• Generalized form of network equilibrium equations for circuits having sources

Branch Current and Loop Current:

Let IB be the branch currents vector and IL be the loop currents vector. BT is the transpose matrix of the fundamental loop matrix or tie set matrix. Then IB branch current can be written as follows

IB = BT.IL

Nothing is new. Actually we have to analyse the network. In actual network if there is a current source in parallel with a passive element let's consider a resistor for simplicity. Then the total branch current will either be the sum or difference of the two currents.

• If current source and resistor current is in same direction, then branch current is the difference of two currents. IB = IS - IR
• If the current source and resistor current is in opposite direction, then branch current is the sum of two currents. IB = IS + IR

In this case the relationship between branch current and loop current can be modified as follows

IB = BT.IL + IS

Where,

Is is the column matrix of order bx1 representing current source across each branch.

Branch Voltage and Node Voltage:

VB = QTVn

Where

VB = branch voltage matrix

QT = transpose of fundamental cut set matrix

Vn = node voltage matrix

Let's consider, a voltage source is present in series branch, then total voltage across this branch will be algebraic sum of source voltage and branch voltage.

• If voltage source in series with a branch with similar polarity then total branch voltage will be the sum of two voltages. V = VS + Vb
• If voltage source in series with a branch with opposite polarity then total branch voltage will be the difference of two voltages. V = VS - Vb

So the above relation between node and branch voltages can be modified as

VB = QT Vn + VS

Where,

Vs = column matrix of order bx1 representing voltage sources in series with each branch

Generalized Form of Network Equilibrium Equations:

I am not going to formulate the network equilibrium equations, I am providing reference books, from where you can get proof of these equations.

Node Equation

YVn = A[YbVs - Is]

The above equation represents (n-1) node equations

Where

Y = AYbAT  is the nodal admittance matrix. This is (n-1) X (n-1) matrix.

A = Reduced incidence matrix

Yb = branch impedance matrix, order (bxb)

Loop Equation

ZIL = B[ZbIs - VS]

The above equation represents (b+n-1) loop equations

E = Z IL

Where,

E = B(VS - ZBIS)

Z = BZbBT is the loop impedance matrix of order (b-n+1)X(b-n+1) matrix

B = fundamental loop or tie set matrix

Zb = is the branch impedance matrix, order (bxb)

E = column matrix of order (b+n-1)x1 representing loop emf

Cut-Set Equation

YCVt = QC[YbVS - IS ]

The above equation represents (n-1) cut-set equations.

J = YC Vt

Where

J = QC(IS - YbVS)

YC = QCYbQCT , is Cut-Set impedance matrix of order (n-1) X (n-1)

QC = fundamental cut-set matrix

Vt = tree voltages

References:

1. Electrical Network Analysis and Synthesis
   By U.A.Bakshi, A.V.Bakshi

Network Analysis And Synthesis
By J.S.Chitode Dr.R.M.Jalnekar

Fundamental Cut Set Matrix - Electrical Network Graphs

This is my fourth article on an electrical network graph. The article is about the cut-set matrix and how it is useful in determining branch voltages and node voltages. As the name implies cut set is a set of branches that when removed from the graph results in an unconnected graph.

Key Questions:

• What is a cut set and a fundamental cut set?
• Procedure for obtaining cut sets of a graph and writing matrix elements
• Fundamental cut set matrix and KVL
• Fundamental cut set and KCL

Cut Set and Fundamental Cut Set:

Cut set: It is a set of branches that when cut or removed from a graph, separates the graph into two groups of nodes or the graph splits into two isolated graphs. It is a minimum set of branches of a connected graph, such that if this set of branches is removed, it will reduce the rank of the graph. And also results in an unconnected graph.

• Cut set defines the minimum set of branches C, if this set of branches removed from the graph results in an unconnected graph.

• If anyone branch from a set of branches C is removed, we still get a connected graph.

Fundamental cut set: It contains only one twig and one or more links.

Procedure For Obtaining Fundamental Cut Set:

As discussed above fundamental cut set contains only one twig and one or more links. To obtain a fundamental cut set follow these steps.

• Select an oriented graph
• Select a tree and mark the links as well with the dotted line
• The number of twigs is equal to the number of cut sets
• Remove one twig and necessary links to get the fundamental cut set
• Repeat the above procedure for all twigs

Fig 1 graph and its tree

Fig 2 Fundamental Cut Sets

Have a closer look at fig 2.

• F-cut set C1 is obtained by removing a twig 5 and a link 3. The direction or orientation of C1 is in the same direction of twig 5
• F-cut set C2 is obtained by removing twig 6 and links 4 and 3. The orientation of C2 is the same as the direction of twig 6
• F-cut set 3 C3 is obtained by removing twig 1 and links 2 & 4. The orientation of C3 is the same as the direction of twig 1

Fundamental Cut Set Matrix (Q):

It describes the branches contained in the cut set and their orientation.

In the cut-set matrix, each row represents a cut set and columns represent branch voltages.

Order of Fundamental Cut Set Matrix:

The number of fundamental cut sets is equal to the number of twigs that is N-1.

B is the total number of branches.

Order of fundamental cut matrix becomes (N-1)xB

Elements of Fundamental Cut Set:

Suppose the graph has 'b’ branches, N nodes, (N-1) twigs in a tree and (N,-1) fundamental cut sets. Let 'i’ be the rows and 'j’ be the columns of the cut-set matrix.

• aij = +1 for the twig of selected fundamental cut set
• aij = +1 if the links have the same current direction as that of the selected twig
• aij = -1 if the links have opposite current direction as that of selected twig
• aij = 0 if the twigs and  are not a part of the selected fundamental cut set

Procedure For Writing Fundamental Cut Set Matrix:

Consider first f-cut set C1

• Branch 5 (twig) = +1
• Branch 3 (link) = -1

Consider second f-cut set C2

• Branch 6 (twig) = +1
• Branch 4 (link) = -1
• Branch 3 (link) = +1

Consider third f-cut set C3

• Branch 1 (twig) = +1
• Branch 2 (link) = +1
• Branch 4 (link) = +1

Fig 3  Fundamental Cut Set Matrix

Fundamental Cut Set Matrix and KCL

Q.Ib = 0

Where Q = cut set matrix

           Ib = branch current vector

Fig 4 KCL and Cut Set Matrix

          -i3 + i5 = 0

          +i3 - i4 + i6 = 0

          i1 + i2 + i4 = 0  

Fundamental Cut Set Matrix and KVL

Vb = QT.Vt

Where Vb = branch voltage vector

          QT = transpose of the cut-set matrix

          Vt = tree branch voltage vector

Fig 5 KVL and Cut Set Matrix

v1 = vt6

v2 = vt6

v3 = vt5 - vt1

v4 = vt6 - vt5

v5 = vt1

v6 = vt5

Tie Set Matrix | Fundamental Loop Matrix - Electrical Network Graph

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This is my third article on electrical network graphs. In this article I am going to explain tie set matrix. Tie set matrix is also known as fundamental loop matrix, circuit matrix, and is denoted by letter B. It is  the set of branches that gives us a closed path. It helps us to determine the branch currents.

Key Questions:

• Explain tie set or fundamental loop
• Explain tie set matrix
• Procedure for writing tie set matrix
• Evaluate branch voltages with the help of tie set matrix using KVL
• Evaluate branch currents with the help of tie set matrix using KCL

Tie Set Matrix And Loop Currents:

Draw Graph o the above circuit.

Tie Set:  Tie set is a set of branches contained in a loop. Each loop consists of a tree and a link or chord.

As you know a tree doesn't contain any closed path or loop. The link is added in a tree in order to join two nodes to form a closed path. This closed path is called fundamental loop. See Fig 1.

Fig 1 Graph, tree and tie sets

Fig 2 Tie set matrix for the fundamental loops in Fig 1

Nodes = N = 4

Branches (twigs) = B = N-1 = 3

Links (chords) = B-(N-1) = 3

• There is a graph
• We select a tree from the given graph
• Now mark all possible fundamental loops
• In order to draw fundamental loop select a link such that it forms a closed path
• Every link defines a fundamental loop
• By convention the direction of current in the fundamental loop is same as that of the link current direction
• The number of fundamental loops is equal to the number of links in cotree. In the above graph there are three fundamental loops

Tie Set Matrix: Tie set matrix is a two dimensional array in which loop currents and branch currents are tabulated. It is the matrix that helps to find out the branch currents.

The row in tie set matrix refers to the fundamental loop currents.

The column in tie set matrix refers to the direction of current in respective branches.

Fundamental loop 1 l1:

Fig 3(a) Fundamental loop 1

• By convention the direction of current in f-loop is same as the direction of current in the link (dotted line) that is branch 3
• Write down the tie set matrix ( row 1) for fundamental loop 1
• Check branch current direction, if it is in the same direction as that of fundamental loop current direction then write +1 for that branch.
• If the branch current is in the opposite direction as that as fundamental loop current direction then write -1 for that branch
• Write 0 if the branch is not the part f fundamental loop
• Branch 1 and branch 5 currents are in the same direction as f-loop while branch 6 current is in opposite direction

Fundamental loop 2 l2:

Fig 3(b) Fundamental loop 2

• First mark the f-loop 2 direction of current. The direction of current in f-loop 2 is same as the direction of current in the link (dotted line) that is branch 2
• Check branch current directions. In branch 1 current is in opposite direction while in branch 2 current is in the same direction as f-loop 2

Fundamental loop 3 l3:

Fig 3(c) Fundamental loop 3

• Loop current is in the direction of link current
• Check branch currents branch 1 and 5 currents are in the same direction as that of f-loop 3

Tie Set Matrix and KVL:

• Each row corresponds to KVL equations, written for fundamental loop. Fundamental loop is added to the tree to form a closed loop
• So there are N-1 KVL equations
• Each column corresponds to branch currents in terms of link currents

v1+v3+v5-v6 = 0

-v1+v2 = 0

v1+v4+v5= 0

BVb=0

Where B is the tie set matrix, Vb is the branch voltage vector.

Fig 4 BVb=0

Tie Set Matrix and KCL:

Fig 5 BT.IL

 i1=I1-I2I1+I3

i2=I2

i3=I1

i4=I3

i5=I1+I3

i6=-I1

IB = BT.IL

Where IB  = branch current vector

IL = loop current

BT = transpose of tie set matrix

Conclusion:

I cover this topic in all aspects. I tried to make this topic as simple as possible, easy to understand and clear. I rechecked all the equations and calculations, but there's always probability of mistakes. So if you find any mistake, please comment. If you have any question you can ask freely. I know this is such a tedious method for analysing a network. It usually helpful for complex networks.