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