⚡ Kirchhoff's Laws — Circuit Analysis Simulator

A two-loop circuit can be solved using either mesh analysis or nodal analysis. With mesh analysis, assign mesh currents I1 and I2 to each loop, then apply KVL: for loop 1: V1 = I1·R1 + (I1−I2)·R2; for loop 2: 0 = I2·R3 + (I2−I1)·R2. Rearrange into matrix form and solve by Gaussian elimination. Nodal analysis finds the voltage at the internal node using KCL, then computes each branch current. This simulator uses MNA, which handles both methods automatically through its matrix formulation.

A Wheatstone bridge has four resistors in a diamond pattern, with a voltage source across one diagonal and a meter across the other. It is balanced when R1/R2 = R3/R4, equivalently R1·R4 = R2·R3 — at this condition no current flows through the galvanometer branch and the bridge voltage is exactly zero. The balance condition allows one unknown resistor to be determined from three known ones. Wheatstone bridges are used in strain gauges, thermistors, and precision instruments achieving accuracies better than 0.1%.

Nodal analysis (based on KCL) solves for node voltages as primary unknowns, writing one equation per non-reference node. It is efficient when there are many nodes but few voltage sources. Mesh analysis (based on KVL) solves for mesh currents, writing one equation per independent loop. It is efficient when there are many loops but few current sources. For planar circuits the number of mesh equations equals B − N + 1 (branches minus nodes plus one). MNA combines both into a unified matrix formulation used by SPICE-based simulators.

KVL applies equally to non-planar circuits — the voltage sum around any closed loop is always zero, regardless of whether the circuit can be drawn without crossing wires. Non-planar circuits cannot use mesh analysis directly, since planar meshes cannot be identified. Instead, nodal analysis or MNA must be used, as KCL at each node is purely topological and does not depend on planarity. The minimum number of independent KVL equations for any circuit is B − N + 1, regardless of planarity.

A supernode is formed when an ideal voltage source connects two non-reference nodes. Since the voltage source enforces a fixed voltage difference between its terminals, the current through it is unknown — a KCL equation using conductance alone cannot be written. The solution is to treat both nodes as a single combined supernode: write KCL for the combined entity (all external currents = 0) and add the voltage constraint V_a − V_b = V_source. This gives a complete set of independent equations. MNA handles supernodes automatically via the B and C sub-matrices.

Dependent (controlled) sources add terms that reference other circuit variables. A Voltage-Controlled Current Source (VCCS) with transconductance g_m injects current g_m·V_ctrl, adding off-diagonal entries to the G matrix. A Current-Controlled Voltage Source (CCVS) adds a new unknown (its current) and a new equation (its voltage constraint). In MNA, all four types — VCCS, VCVS, CCCS, CCVS — are incorporated by extending the B, C, D sub-matrices, preserving the sparse structure needed for efficient simulation.

The superposition theorem states that in a linear circuit with multiple independent sources, the response at any element equals the sum of responses produced by each source acting alone, with all other sources replaced by their internal impedances (voltage sources → short circuits, current sources → open circuits). Superposition simplifies analysis when sources have different frequencies (e.g., AC + DC mixed circuits). It does not apply to power calculations since power is nonlinear (P = I²R), nor to circuits with nonlinear elements such as diodes or transistors.