A voltage node is typically a conjunction in an electric circuit at which an electric potential can be calculated with respect to some other (reference) node. if one point in the circuit is grounded, that point is generally selected as the reference node. Otherwise, any convenient junction can be treated as a reference node.

Determine the voltage at each node of the circuit of figure:

FROM ABOVE CIRCUIT, we can write the following set of equations:

$\begin{align} & {{I}_{1}}={{I}_{{{R}_{1}}}}+{{I}_{{{R}_{3}}}} \\ & {{I}_{1}}={{I}_{{{R}_{2}}}}+{{I}_{{{R}_{4}}}} \\ & {{V}_{1}}={{V}_{A}}-{{V}_{C}} \\\end{align}$

So

\[\begin{matrix} {{I}_{1}}=\frac{{{V}_{B}}-{{V}_{A}}}{{{R}_{1}}}+\frac{{{V}_{B}}-{{V}_{C}}}{{{R}_{3}}} & {} & {} \\ {{I}_{1}}=\frac{{{V}_{A}}}{{{R}_{2}}}+\frac{{{V}_{C}}}{{{R}_{4}}} & {} & (1) \\ {{V}_{1}}={{V}_{A}}-{{V}_{C}} & {} & {} \\\end{matrix}\]

THE ABOVE EQUATIONS CAN BE WRITTEN IN TERMS OF matrix:

$[A]=[B]*[C]$

Whereas:

$[A]=\left[ \begin{matrix} {{I}_{1}} \\ {{I}_{1}} \\ {{V}_{1}} \\\end{matrix} \right]$

\[[B]=\left[ \begin{matrix} -\frac{1}{{{R}_{1}}} & \frac{1}{{{R}_{3}}}+\frac{1}{{{R}_{1}}} & -\frac{1}{{{R}_{3}}} \\ \frac{1}{{{R}_{2}}} & 0 & \frac{1}{{{R}_{4}}} \\ 1 & 0 & -1 \\\end{matrix} \right]\]

$[C]=\left[ \begin{matrix} {{V}_{A}} \\ {{V}_{B}} \\ {{V}_{C}} \\\end{matrix} \right]$

So, by using the following equation, we can easily determine the values of unknowns:

$[C]={{[B]}^{-1}}[A]$

Now, let’s put up all above equations in Matlab and see what we get:

**Matlab Code for Node Voltages Calculation in a Bridge Circuit**

%Bridge Circuit clear all;close all;clc V1= 2; % Source Voltage I1= 0.2; % Source Current R_1= 80; R_2= 70; % Resistances in the Circuit R_3= 60; R_4= 90; A=[I1;I1;V1]; % Input Matrix from equation (1) B=[-1/R_1 1/R_1+1/R_3 -1/R_3; ... 1/R_2 0 1/R_4 ; ... % Elements of VA, VB, and VC in equation (1) 1 0 -1 ]; C=inv(B)*A; VA=C(1,1) VB=C(2,1) % Unknown Node Voltages VC=C(3,1) %==============================================

** Results:**

VA =

8.7500

VB =

14.4643

VC =

6.7500