Nodal Analysis Calculator: Three-Node Circuits, Node Voltages and Branch Currents

Solve circuits by the node voltage method with Kirchhoff's current law.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter, for each node, the source voltage and its series resistor (0 for none) and the resistor to ground.

Enter the resistors between the nodes. Use 0 where there is none.

The calculator writes the node equations, solves them, and gives the node voltages and all the currents, with a check.

Formulas used

  • [G][V] = [I]
  • G_{kk} = \text{sum of conductances at node }k
  • G_{jk} = -\frac{1}{R_{jk}}
  • I_k = \frac{V_{s,k}}{R_{s,k}}

Worked example

With the default numbers in the calculator above:

  1. Node equations (Kirchhoff's current law at each node): the conductance 1/R on the diagonal, minus the conductance between nodes off it: Node 1: 1.2 V1 - 0.5 V2 - 0 V3 = 5 A; Node 2: -0.5 V1 + 1.0833 V2 - 0.33333 V3 = 0 A; Node 3: -0 V1 - 0.33333 V2 + 1.5 V3 = 5 A (conductances in siemens)
  2. Node voltages (solved by Gaussian elimination), measured from ground: 5.83 V at node 1, 3.99 V at node 2, 4.22 V at node 3
  3. Current from each source into its node: source 1 gives 2.09 A into node 1; source 3 gives 0.78 A into node 3
  4. Current in each resistor (positive from the first node named toward the second or toward ground): ground at node 1: 1.17 A (5.83 V across); ground at node 2: 1.00 A (3.99 V across); ground at node 3: 0.70 A (4.22 V across); between nodes 1 and 2: 0.92 A (1.84 V across); between nodes 2 and 3: -0.0771 A (-0.23 V across)
  5. Check: Kirchhoff's current law at node 1: The currents at node 1 add to 0 A (zero)
  6. Power supplied by the sources and power lost in the resistors: 24.76 W supplied and 24.76 W in the resistors (equal)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is nodal analysis?

A method that writes Kirchhoff's current law at each node, with the node voltages as the unknowns.

When is nodal better than mesh analysis?

When there are fewer nodes than loops, which is common in circuits with many parallel branches.

What is the reference node?

The ground node, chosen as zero volts. All node voltages are measured from it.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year electrical circuits text (nodal analysis).

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