Example 1

Nerd Cafe

To solve the circuit in Fig. below using Nodal Analysis, we will calculate the voltage at each node step by step. Let’s start by identifying the steps and equations involved.

1. Steps for Nodal Analysis:

Assign Node Voltages:

• Label the node voltages as 𝑉₁, 𝑉₂, 𝑉₃, 𝑉₄ and 𝑉₀ (ground node).

• 𝑉₀ is already set to 0 V as the reference.

• 𝑉₁ is equal to 8 V.

• 𝑉₄ is equal to (-8) V.

Apply Kirchhoff's Current Law (KCL):

• At each node, the sum of currents leaving the node equals zero.

Write Equations for Each Node:

• Using Ohm’s Law (𝐼=𝑉/𝑅).

2. Solve the System of Equations:

• Use substitution or matrix methods to find the node voltages.

3. Circuit Breakdown

Known Values:

• Voltage sources: 𝑉₁=8V, 𝑉₂=8V.

• Resistances: 𝑅₁=4Ω, 𝑅₂=6Ω, 𝑅₃=7Ω, 𝑅₄=9Ω, 𝑅₅=10Ω.

• Current sources: 𝐼₁=5A, 𝐼₂=6A.

4. KCL Equations

Node 1 (𝑉₁):

$$V_{1}=8$$

Node 2 (𝑉₂):

$$\frac{V_{2}-8}{4}-5+\frac{V_{2}}{6}+\frac{V_{2}-V_{3}}{7}=0$$

Node 3 (𝑉₃):

$$\frac{V_{3}-V_{2}}{7}+\frac{V_{3}-(-8)}{9}+\frac{V_{3}}{10}+6=0$$

Node 4 (V₄):

$$V_{4}=-8$$

4. Simplified Equations:

Node 2 (Simplified):

$$V_{2}\left( \frac{1}{4}+\frac{1}{+}+\frac{1}{7} \right)+V_{3}\left( -\frac{1}{7} \right)=7\Rightarrow V_{2}\left( 0.5595238 \right)+V_{3}\left( -0.142857 \right)=7$$

5. Matrix Form

The equations can be expressed in matrix form as:

$$\begin{bmatrix} 0.5595238 & -0.142857 \\ -0.142857 & 0.353968 \end{bmatrix}\begin{bmatrix} V_{2} \\ V_{3} \end{bmatrix}=\begin{bmatrix} 7 \\ -6.8888 \end{bmatrix}$$

6. Python code

Here’s the Python code to solve the node voltages using NumPy for matrix operations:

import numpy as np

# Coefficient matrix
A = np.array([
    [0.5595238, -0.142857],
    [-0.142857, 0.353968]
])

# Constant matrix
B = np.array([7, -6.8888])

# Solve the linear system Ax = B
solution = np.linalg.solve(A, B)

# Print the results
V2, V3 = solution
print(f"V2 = {V2}")
print(f"V3 = {V3}")

Output

V2 = 8.408113665230585
V3 = -16.06823810662589

Keywords

nodal analysis, kirchhoff's current law, KCL, circuit analysis, node voltage method, electrical engineering, electronic circuits, DC circuits, AC circuits, linear circuits, circuit solving techniques, voltage nodes, current sources, resistive circuits, mesh analysis, supernode, circuit theory, circuit equations, Ohm's law, engineering education