Determine the voltage at each node of the circuit:
FROM ABOVE CIRCUIT, we can write the following set of equations:
$\begin{matrix} \begin{align} & {{I}_{{{R}_{4}}}}={{I}_{{{R}_{1}}}}+{{I}_{{{R}_{3}}}} \\ & {{I}_{{{R}_{2}}}}={{I}_{{{R}_{3}}}} \\ & {{V}_{1}}={{V}_{A}} \\\end{align} & \cdots & (1) \\\end{matrix}$
Or
\[\begin{matrix} \begin{align} & \frac{{{V}_{C}}}{{{R}_{4}}}=\frac{{{V}_{1}}-{{V}_{C}}}{{{R}_{1}}}+\frac{{{V}_{B}}-{{V}_{C}}}{{{R}_{3}}} \\ & \frac{{{V}_{1}}-{{V}_{B}}}{{{R}_{2}}}=\frac{{{V}_{B}}-{{V}_{C}}}{{{R}_{3}}} \\\end{align} & \cdots & (2) \\\end{matrix}\]
We can rewrite it as:
\[\begin{matrix} \begin{align} & \frac{{{V}_{1}}}{{{R}_{1}}}=\left( \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{3}}}+\frac{1}{{{R}_{4}}} \right){{V}_{C}}-\frac{{{V}_{B}}}{{{R}_{3}}} \\ & \frac{{{V}_{1}}}{{{R}_{2}}}=-\frac{{{V}_{C}}}{{{R}_{3}}}+\left( \frac{1}{{{R}_{3}}}+\frac{1}{{{R}_{2}}} \right){{V}_{B}} \\\end{align} & \cdots & (3) \\\end{matrix}\]
The above equations can be written in terms of matrix:
$[A]=[B]*[C]$
Whereas:
$[A]=\left[ \begin{matrix} \frac{{{V}_{1}}}{{{R}_{1}}} \\ \frac{{{V}_{1}}}{{{R}_{2}}} \\\end{matrix} \right]$
$[B]=\left[ \begin{matrix} \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{3}}}+\frac{1}{{{R}_{4}}} & -\frac{1}{{{R}_{3}}} \\ -\frac{1}{{{R}_{3}}} & \frac{1}{{{R}_{3}}}+\frac{1}{{{R}_{2}}} \\\end{matrix} \right]$
$[C]=\left[ \begin{matrix} {{V}_{C}} \\ {{V}_{B}} \\\end{matrix} \right]$
By using following equation, we can obtain unknown variables:
$[C]={{[B]}^{-1}}[A]$
Let’s calculate unknown elements using Matlab:
- You May Also Read: Nodal Analysis Theory
Matlab Code for Node Voltages Calculation
%Simple Resistive Network clear all;close all;clc %%Circuit Elements V1= 2; % Source Voltage R1= 330; R2= 70; % Resistances in Ohm from Circuit R3= 160; R4= 270; B=[1/R1+1/R3+1/R4 -1/R3 ; ... -1/R3 1/R3+1/R2]; % Elements of V1, VB, and VC in equation (3) A=[V1/R1; V1/R2]; % Inputs Vector C=inv(B)*A; % Output Matrix % Node Voltages V_C=C(1,1) V_B=C(2,1) %============================================================
Results:
V_C =
1.3316
V_B =
1.7966
