In this tutorial, we will find node voltages for a very simple resistive circuit using Nodal Analysis.
While applying KCL, we will assume that currents leaving the node are positive and entering the node are negative. Keeping that fact in mind, let’s write node voltages for each node in the circuit.
Node 1:
\[\begin{matrix} \begin{align} & \frac{{{V}_{1}}-{{V}_{2}}}{10}+\frac{{{V}_{1}}-{{V}_{3}}}{10}-5=0 \\ & 0.15{{V}_{1}}-0.1{{V}_{2}}-0.05{{V}_{3}}=5 \\\end{align} & \cdots & (1) \\\end{matrix}\]
Node 2:
\[\begin{matrix} \begin{align} & \frac{{{V}_{2}}-{{V}_{1}}}{10}+\frac{{{V}_{2}}}{50}+\frac{{{V}_{2}}-{{V}_{3}}}{40}=0 \\ & -0.10{{V}_{1}}+0.145{{V}_{2}}-0.025{{V}_{3}}=0 \\\end{align} & \cdots & (2) \\\end{matrix}\]
Node 3:
\[\begin{matrix} \begin{align} & \frac{{{V}_{3}}-{{V}_{1}}}{20}+\frac{{{V}_{3}}-{{V}_{2}}}{40}-2=0 \\ & -0.05{{V}_{1}}-0.025{{V}_{2}}+0.075{{V}_{3}}=2 \\\end{align} & \cdots & (3) \\\end{matrix}\]
Let’s combine all (1), (2), and (3) in Matrix form
$\left[ \begin{matrix} 0.15 & -0.1 & -0.05 \\ -0.1 & 0.145 & -0.025 \\ -0.05 & -0.025 & 0.075 \\\end{matrix} \right]\left[ \begin{matrix} {{V}_{1}} \\ {{V}_{2}} \\ {{V}_{3}} \\\end{matrix} \right]=\left[ \begin{matrix} 5 \\ 0 \\ 2 \\\end{matrix} \right]$
- You May Also Read: Nodal Analysis Theory
Now, we will write a small piece of Matlab code to compute all node voltages.
clear all;close all;clc % Nodal Analysis using Matlab Y_Mat = [ 0.15 -0.1 -0.05; -0.1 0.145 -0.025; % Admittance Matrix (YV=I) obtain from Node equations -0.05 -0.025 0.075]; I_vec = [5; 0; % Current Vector (again from Node equations) 2]; %% Node Voltages Calculation fprintf('Nodal voltages V1, V2 and V3 are \n') V_Node = inv(Y_Mat)*I_vec
Results:
Nodal voltages V1, V2, and V3 are
V_Node =
404.2857
350.0000
412.8571