Matrix Equation Builder
Description
This MicroSim takes a simple resistive circuit as input and assembles the full matrix equation \(\mathbf{G}\mathbf{V} = \mathbf{I}\) for nodal analysis (or \(\mathbf{Z}\mathbf{I} = \mathbf{V}\) for mesh analysis). Each matrix entry is color-coded to show which element contributed it, and the solution vector is computed and displayed. Toggle between nodal and mesh modes to see the duality in action.
Key Concepts
- Nodal analysis matrix form: \(\mathbf{G}\mathbf{V} = \mathbf{I}_s\), where \(\mathbf{G}\) is the conductance matrix assembled by inspection.
- Diagonal entries \(G_{kk}\) = sum of conductances connected to node k.
- Off-diagonal entries \(G_{kj}\) = negative sum of conductances between nodes k and j.
- Mesh analysis matrix form: \(\mathbf{Z}\mathbf{I}_m = \mathbf{V}_s\), with the impedance matrix assembled by similar inspection rules.
- The matrix equation can be solved by Gaussian elimination, Cramer's rule, or matrix inversion \(\mathbf{V} = \mathbf{G}^{-1}\mathbf{I}_s\).