Two-node voltage analysis calculator

Enter the conductance from each node to ground, the conductance connecting the two nodes, and the net current entering each node. The calculator builds the simultaneous nodal equations and returns both voltages relative to ground. It is useful for checking hand calculations, validating a compact circuit model, or adding a deterministic two-node solve to an engineering workflow without rearranging the equations yourself.

Describe the two-node circuit in conductance form

Represent the network with three conductances measured in siemens: one from node 1 to ground, one from node 2 to ground, and one between the two nodes. Conductance is the reciprocal of resistance, so a 1 kΩ branch becomes 0.001 S and a 2 kΩ branch becomes 0.0005 S. Then provide the net injected current at each node in amperes. Use a positive sign when a source pushes current into the node and a negative sign when the specified current leaves it. Keeping that sign convention consistent is essential because nodal analysis is an application of Kirchhoff’s current law. A branch may have zero conductance when it is absent. However, the overall network must include a conductive reference to ground; otherwise, both absolute node voltages can float together and the equations do not have a unique solution. The input fields describe topology explicitly, which avoids ambiguity about whether a given resistor is grounded or connects the two unknown nodes. Convert every branch to siemens before entering it, and use one consistent current unit throughout.

Understand the simultaneous equations

The calculator constructs a 2×2 conductance matrix. For node 1, the diagonal coefficient is the sum of the conductances touching node 1, while the coefficient multiplying node 2 is the negative of the conductance between the nodes. Node 2 follows the same pattern. In equation form, (G1 + G12)V1 − G12V2 = I1 and −G12V1 + (G2 + G12)V2 = I2. The solver evaluates the determinant and applies the closed-form solution for this linear system. A positive determinant gives a unique pair of voltages for a passive network with non-negative branch conductances. A zero determinant means the conductance matrix is singular, usually because neither node has an effective path to the ground reference. That condition is reported as invalid input instead of returning arbitrary or infinite voltages. The returned determinant is included as a useful diagnostic: it confirms what matrix was solved and can reveal a circuit that is approaching a poorly conditioned configuration, although it is not by itself a complete condition-number analysis.

Check units, signs, and the resulting voltages

Read the two voltage results as potentials relative to ground. With conductance in siemens and current in amperes, the result is in volts because the nodal system follows I = GV. Negative voltage is valid and does not indicate a solver error; it can arise whenever the net source directions pull a node below the chosen ground reference. For a quick verification, substitute both returned voltages into the original node equations. The calculated branch-current sums should reproduce the two injected currents, allowing for ordinary floating-point rounding. Also consider limiting cases. If the inter-node conductance is zero, each node should reduce to its independent current-over-ground-conductance result. If the inter-node conductance becomes large, the two voltages should move closer together. This capability is intended for linear, resistive, two-node models. Convert voltage sources with series resistance to equivalent Norton current sources and conductances before use. Do not apply the result directly to capacitors, inductors, nonlinear semiconductor models, or time-varying behavior unless those elements have already been replaced by a valid linear conductance model for the specific analysis point.