Another alternate method of analysis of multiple loop circuits is called the node voltage method. It is based on finding the voltages at each node in the circuit using Kirchhoff s current law. A node is the junction of two or more current paths.
The general steps for the node voltage method are as follows:
We will use Figure-1. to ilustrate the general approach to node voltage analysis.
Figure - 1
Establish the node. In this case there are four nodes, as indicated in the figure.
|Second||Let's use node B as reference. Think of it as
circuit ground. Node
voltages C and D are already known to he the source voltages. voltage's. The voltage at node A is he only unknown it is designated as VA.
|Third||Arbitrarily assign the currents at a node A as indicated in the figure-1|
The Kirchhoff's current equation at node A is
I1 - I2 + I3 = 0
Express the currents in terms of circuit voltages using Ohm's law as follow
I1 = V1/R1 = (VS1 - VA)/R1
I2 = V2/R2 = VA/R2
I3 = V3/R3 = (VS2 - VA)/R3
Substituting these into the current equation, we get
(VS1 - VA)/R1 -VA/R2 + (VS2 - VA)/R3 = 0
The only unknown is VA; so we can solve the single equation by combining and rearranging terms Once the voltage is know, all branch currents can be calculated.
Find the node voltages in Figure
The unknown node voltage is VA, as indicated in figure. This is the only unknown voltage. Currents are assigned at node A as shown. The current equation is
I1 - I2 + I3 = 0
Substitution for current using Ohm's law gives equation in terms of voltages
(10 -VA)/47 - VA/22 + (5 - VA)/82 = 0
Solving for VA yields
10/47 - VA/47 - VA/22 + 5/82 - VA/82 = 0
- VA/47 - VA/22 - VA/82 = - 10/47 - 5/82
(1804VA + 3854VA + 1034VA)/84788 = (820 + 235)/3854
6692VA / 84788 = 1055/3854
VA = (1055 X 84788)/(6692 X 3854) = 3.47 V
VA = 3.47 volt