An equally potent substitute for mesh current analysis is nodal voltage analysis. Matrix analysis is used in both techniques to provide effective circuit solutions. Nodal Voltage Analysis, as the name implies, uses nodal equations to represent Kirchhoff’s Current Law (KCL) and determines the voltage potentials around the circuit.
The fundamental idea is that, for any given node, the total current flowing into and out of that node must be equal (KCL). By doing this for every node (apart from a selected reference node), we have a set of “n-1” independent equations that describe the circuit as a whole, where n is the number of nodes.
All of these equations may be solved by expressing each current in terms of the voltage difference (potential) across the branch to which it belongs. In the end, out of n nodes, one is assigned as the reference, having a voltage of zero. The voltages of the other n-1 nodes are then calculated in relation to this reference.
To illustrate Nodal Voltage Analysis, let’s revisit the circuit from the last section.
Understanding Nodal Voltage Analysis through a Circuit
Here we designate node D as the reference node, meaning that its voltage is set to zero (0 V), in order to simplify the analysis.
The potential differences between each node and the reference node D are represented by the voltages at the remaining nodes, which are designated as Va, Vb, and Vc.
- (Va – Vb) / 9 + (Vc – Vb) / 18 = Vb / 36
- Given that Va = 9 volts and Vc = 18 volts, finding Vb is simple.
- (1 – Vb / 9) + (1 – Vb / 18) = Vb / 36
- First, let’s simplify the equation:
- 9 / 9 − 9 / Vb + 18 / 18 − 18 / Vb = 36 / Vb
- Combining the fractions:
- 1 − 9 / Vb + 1 −18 / Vb = 36 / Vb
- Now, let’s add the terms on the left side:
- 2 − 9 / Vb − 18 / Vb = 36 / Vb
- To get rid of the fractions, let’s multiply both sides of the equation by 36:
- 36 * 2 − 36 * Vb / 9 − 36 * Vb / 18 = Vb
- Solving for Vb:
- 72 − 4Vb − 2Vb = Vb
- Combining like terms:
- 72 − 6Vb = Vb
- Adding 6Vb to both sides:
- 72 = 7Vb
- Therefore, the value of Vb is = 72/7 V
- Thus, I3 = 2 / 7 = 0.286 amps
The answer is once more the exact same value of 0.286 amps that we discovered in the previous lesson utilizing Kirchhoff’s Circuit Law.
Nodal Voltage Analysis: An Efficient Option
Nodal Voltage Analysis turns out to be a more straightforward approach than Mesh Current Analysis for the circuit we looked at. Generally speaking, circuits with more current sources perform better when using Nodal Voltage Analysis.
It is mainly because a mathematical framework is used in nodal analysis. Kirchhoff’s Current Law, which describes the relationship between currents entering and exiting each node, may be expressed as a matrix equation.
This formula appears as follows:
[I] = [Y] [V]
[I]: This matrix shows the circuit’s driving current sources.
[V]: This matrix indicates the unknown nodal voltages that we wish to solve for.
[Y]: This matrix, also known as the admittance matrix, records the circuit components’ conductance data. We may get the current source matrix by multiplying the admittance matrix by the nodal voltage matrix.
This matrix formulation makes it possible to solve the circuit effectively, especially when there are more current sources available.
Conclusions:
The main procedures for resolving circuits with Nodal Analysis are outlined in this summary:
Define the current vectors for each of the circuit’s “N” independent nodes. It is customary to view currents entering a node as beneficial. N rows (N x 1) of a column matrix will be created using these vectors.
The Admittance Matrix Design: Create a N x N square matrix that will be referred to as the admittance matrix ([Y]). This matrix records the circuit’s conductance information, as given below:
- The total admittance associated with each node is represented by the diagonal elements (Y11, Y22, etc.).
- The total admittance shared directly between nodes J and K is represented by off-diagonal elements (Yjk); these values might be zero or negative.
Voltage Vector: In order to solve for the unknown nodal voltages ([V]), build a column matrix (N x 1).
We have looked at how circuit analysis is made simpler by Nodal Analysis. Thevenin’s Theorem will be covered in detail in the next lesson. With the help of this effective theorem, we can represent a complicated network of resistors and sources with an equivalent circuit that is considerably simpler and only needs a single voltage source and a series resistance!
References: Node voltage method