Network Equilibrium Equation in Matrix Form

Network Equilibrium Equation in Matrix Form:

In earlier section we have studied how to obtain equilibrium equations for a network which are based on loop analysis and node analysis. We can represent the same Network Equilibrium Equation in Matrix Form which simplifies the task of analysis of a large scale network.

Loop or Mesh or KVL Equilibrium Equations:

The KVL equilibrium equations based on loop or mesh analysis are given by

Network Equilibrium Equation in Matrix Form

where

Network Equilibrium Equation in Matrix Form

The quantities involved in above equations are as follows.

B is the f circuit or tie-set matrix of order (b – n+1) x b,

Zb is the branch impedance matrix of order b x b,

IL is the column matrix of loop currents of order (b – n + 1) x 1

Vs is the column matrix of voltage sources of order b x 1.

Is is column matrix of source currents of order b x 1

Z is coefficient matrix also known as loop impedance matrix of order (b – n +1) x (b – n +1)

E is the column matrix of order (b – n +1) x 1 which represents voltages in loop.

The matrix equation ZIL = E represents a set of (b – n + 1) independent loop equations for a network with n nodes and b branches.

Node or KCL Equilibrium Equation:

The KCL equilibrium equations based on nodal analysis are given by

Network Equilibrium Equation in Matrix Form

where

The quantities involved in above equations are as follows.

Q is the f-cutset matrix of order (n -1) x b

Yb is branch admittance matrix of order b x b

Vt is column matrix of tree branch voltages of order (n -1) x 1

Is is the column matrix of branch input current sources of order b x 1

Vs is column matrix of branch input voltage sources

Y is node admittance matrix of order (n -1) x (n -1)

I is the column matrix of order (n -1) x 1 which represents algebraic sum of current sources.

Updated: November 1, 2019 — 9:16 pm