DC Circuit Solver and Kirchhoff Equations

What does this tool do?

It solves a direct current circuit made of resistors, voltage sources and current sources. It finds the voltage of every node and the current and power of every element, checks the power balance and writes out the Kirchhoff equations of the circuit. Bridge circuits and circuits with several sources, which cannot be reduced by series and parallel rules, are solved as well.

How to enter a circuit

Give every junction (node) in the circuit a name: 0, 1, 2 or a, b, c. Then, for each element, enter the two nodes it connects to and its value. Terminals with the same name are joined.

• Resistor: two nodes and the value in ohms.
• Voltage source: the node of the plus terminal, the node of the minus terminal and the value in volts.
• Current source: the node the current flows into, the node it flows out of and the value in amperes.

One node is chosen as ground (0 V). All other voltages are given relative to it. Changing the ground shifts the voltages but leaves currents and powers unchanged.

Kirchhoff's laws

• Current law (KCL): the currents entering a node equal the currents leaving it.
• Voltage law (KVL): the voltage drops around any closed loop add up to zero.

The tool writes the current law equation for each node and the source equation for each voltage source, then solves the set. It also shows with numbers that the voltage law holds around the loops of the circuit.

Example

A 12 V source feeds a middle node through 4 Ω, a 3 V source is connected to the same node through 2 Ω, and a 4 Ω resistor runs from the middle node to ground. The current law at the middle node gives (x − 12)/4 + (x − 3)/2 + x/4 = 0, so x = 4.5 V. The 12 V source delivers 22.5 W at 1.875 A. A current of 0.75 A flows into the 3 V source, so it absorbs 2.25 W. The resistors dissipate 20.25 W in total and the balance holds.

Method

The calculation uses modified nodal analysis. The unknowns are the node voltages other than ground and the currents of the voltage sources. The resulting linear system is solved by Gaussian elimination.

Limits

The tool works for direct current only, with fixed resistors and ideal independent sources. There are no dependent sources, capacitors, inductors, diodes or transistors. Ideal voltage sources cannot be connected in parallel or have both terminals on one node. A node with no path to ground through a resistor or a voltage source has an undefined voltage. The tool reports each of these cases by name.