Theory — Kirchhoff’s Rules (Matrix Method)
1) Kirchhoff’s Current Law (KCL)
At any node, the algebraic sum of currents is zero:
\( \sum I = 0 \).
If a current comes out negative, the true direction is opposite to your chosen direction.
2) Kirchhoff’s Voltage Law (KVL)
Around any closed loop, the sum of potential changes is zero:
\( \sum \Delta V = 0 \).
For a resistor, \( \Delta V = IR \).
3) Modified Nodal Analysis (MNA)
With ground \(N_0\) and unknown node voltages \(V_1,\dots,V_n\), MNA builds a linear system:
- \(G\): conductance matrix from resistors
- \(B\): incidence of ideal voltage sources
- \(\mathbf{e}\): source voltages
- \(\mathbf{I}_{vs}\): currents through ideal sources (additional unknowns)
Series resistors in one branch
Multiple resistors in series combine as
\( R_{eq} = R_1 + R_2 + \cdots \).
Parallel paths are modeled by adding multiple branches between the same nodes.
