of the two nodes are combined in a single equation, and a new equation for the voltages is formed. Solve the system of simultaneous equations for each unknown voltage. Examples Basic case The only unknown voltage in this circuit is . There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node. Current through resistor : Current through resistor : Current through current source : With Kirchhoff's current law, we get: This equation can be solved with respect to V1: Finally, the unknown voltage can be solved by substituting numerical values for the symbols. Any unknown currents are easy to calculate after all the voltages in the circuit are known. Supernodes In this circuit, we initially have two unknown voltages, V1 and V2. The voltage at V3 is already known to be VB because the other terminal of the voltage source is at ground potential. The current going through voltage source VA cannot be directly calculated. Therefore, we cannot write the current equations for either V1 or V2. However, we know that the same current leaving node V2 must enter node V1. Even though the nodes cannot be individually solved, we know that the combined current of these two nodes is zero. This combining of the two nodes is called the supernode technique, and it requires one additional equation: V1 = V2 + VA. The complete set of equations for this circuit is: By substituting Matrix form for the node-voltage equation In general, for a circuit with nodes, the node-voltage equations obtained by nodal analysis can be written in a matrix form as derived in the following. For any node , KCL states where is the negative of the sum of the conductances between nodes and , and is the voltage of node . This implies where is the sum of conductances connected to node . We note that the first term contributes linearly to the node via , while the second term contributes linearly to each node connected to the node via with a minus sign. If an independent current source/input is also attached to node , the above expression is generalized to . It is readily shown that one can combine the above node-voltage equations for all nodes, and write them down in the following matrix form or simply The matrix